L(s) = 1 | − 12·5-s + 78·25-s + 4·37-s + 8·41-s + 36·43-s − 32·47-s − 4·67-s − 28·79-s − 40·83-s − 40·89-s − 32·101-s + 4·109-s + 64·121-s − 364·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 54·169-s + 173-s + 179-s + 181-s + ⋯ |
L(s) = 1 | − 5.36·5-s + 78/5·25-s + 0.657·37-s + 1.24·41-s + 5.48·43-s − 4.66·47-s − 0.488·67-s − 3.15·79-s − 4.39·83-s − 4.23·89-s − 3.18·101-s + 0.383·109-s + 5.81·121-s − 32.5·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4.15·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 5^{12} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 5^{12} \cdot 7^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.02190469661\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.02190469661\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( ( 1 + T )^{12} \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 64 T^{2} + 2078 T^{4} - 43908 T^{6} + 684943 T^{8} - 8690684 T^{10} + 98817832 T^{12} - 8690684 p^{2} T^{14} + 684943 p^{4} T^{16} - 43908 p^{6} T^{18} + 2078 p^{8} T^{20} - 64 p^{10} T^{22} + p^{12} T^{24} \) |
| 13 | \( 1 - 54 T^{2} + 113 p T^{4} - 27038 T^{6} + 379302 T^{8} - 4418150 T^{10} + 52195829 T^{12} - 4418150 p^{2} T^{14} + 379302 p^{4} T^{16} - 27038 p^{6} T^{18} + 113 p^{9} T^{20} - 54 p^{10} T^{22} + p^{12} T^{24} \) |
| 17 | \( ( 1 + 50 T^{2} - 108 T^{3} + 1061 T^{4} - 5136 T^{5} + 16550 T^{6} - 5136 p T^{7} + 1061 p^{2} T^{8} - 108 p^{3} T^{9} + 50 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 19 | \( 1 - 66 T^{2} + 2789 T^{4} - 89918 T^{6} + 2310498 T^{8} - 51526010 T^{10} + 1031191685 T^{12} - 51526010 p^{2} T^{14} + 2310498 p^{4} T^{16} - 89918 p^{6} T^{18} + 2789 p^{8} T^{20} - 66 p^{10} T^{22} + p^{12} T^{24} \) |
| 23 | \( 1 - 4 p T^{2} + 5158 T^{4} - 205024 T^{6} + 6913435 T^{8} - 198569132 T^{10} + 4980497936 T^{12} - 198569132 p^{2} T^{14} + 6913435 p^{4} T^{16} - 205024 p^{6} T^{18} + 5158 p^{8} T^{20} - 4 p^{11} T^{22} + p^{12} T^{24} \) |
| 29 | \( 1 - 200 T^{2} + 19654 T^{4} - 1282780 T^{6} + 62801839 T^{8} - 2446961612 T^{10} + 78075542648 T^{12} - 2446961612 p^{2} T^{14} + 62801839 p^{4} T^{16} - 1282780 p^{6} T^{18} + 19654 p^{8} T^{20} - 200 p^{10} T^{22} + p^{12} T^{24} \) |
| 31 | \( 1 - 162 T^{2} + 12269 T^{4} - 559814 T^{6} + 16573002 T^{8} - 339587402 T^{10} + 7404552533 T^{12} - 339587402 p^{2} T^{14} + 16573002 p^{4} T^{16} - 559814 p^{6} T^{18} + 12269 p^{8} T^{20} - 162 p^{10} T^{22} + p^{12} T^{24} \) |
| 37 | \( ( 1 - 2 T + 117 T^{2} - 122 T^{3} + 6912 T^{4} - 2818 T^{5} + 291033 T^{6} - 2818 p T^{7} + 6912 p^{2} T^{8} - 122 p^{3} T^{9} + 117 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 41 | \( ( 1 - 4 T + 200 T^{2} - 744 T^{3} + 18359 T^{4} - 57292 T^{5} + 970430 T^{6} - 57292 p T^{7} + 18359 p^{2} T^{8} - 744 p^{3} T^{9} + 200 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 43 | \( ( 1 - 18 T + 233 T^{2} - 1918 T^{3} + 15792 T^{4} - 108898 T^{5} + 797957 T^{6} - 108898 p T^{7} + 15792 p^{2} T^{8} - 1918 p^{3} T^{9} + 233 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 47 | \( ( 1 + 16 T + 242 T^{2} + 2064 T^{3} + 17903 T^{4} + 109312 T^{5} + 836636 T^{6} + 109312 p T^{7} + 17903 p^{2} T^{8} + 2064 p^{3} T^{9} + 242 p^{4} T^{10} + 16 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 53 | \( 1 - 468 T^{2} + 107090 T^{4} - 15743908 T^{6} + 1650122943 T^{8} - 129607263208 T^{10} + 7812909910460 T^{12} - 129607263208 p^{2} T^{14} + 1650122943 p^{4} T^{16} - 15743908 p^{6} T^{18} + 107090 p^{8} T^{20} - 468 p^{10} T^{22} + p^{12} T^{24} \) |
| 59 | \( ( 1 + 188 T^{2} + 228 T^{3} + 16139 T^{4} + 34548 T^{5} + 1003694 T^{6} + 34548 p T^{7} + 16139 p^{2} T^{8} + 228 p^{3} T^{9} + 188 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 61 | \( 1 - 372 T^{2} + 73898 T^{4} - 10176716 T^{6} + 1066215327 T^{8} - 88729193408 T^{10} + 5990506153820 T^{12} - 88729193408 p^{2} T^{14} + 1066215327 p^{4} T^{16} - 10176716 p^{6} T^{18} + 73898 p^{8} T^{20} - 372 p^{10} T^{22} + p^{12} T^{24} \) |
| 67 | \( ( 1 + 2 T + 223 T^{2} + 430 T^{3} + 27322 T^{4} + 47858 T^{5} + 2193629 T^{6} + 47858 p T^{7} + 27322 p^{2} T^{8} + 430 p^{3} T^{9} + 223 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 71 | \( 1 - 396 T^{2} + 82922 T^{4} - 11857216 T^{6} + 1291830615 T^{8} - 115038320068 T^{10} + 8757474588608 T^{12} - 115038320068 p^{2} T^{14} + 1291830615 p^{4} T^{16} - 11857216 p^{6} T^{18} + 82922 p^{8} T^{20} - 396 p^{10} T^{22} + p^{12} T^{24} \) |
| 73 | \( 1 - 778 T^{2} + 283945 T^{4} - 64214018 T^{6} + 10014485938 T^{8} - 1134094607338 T^{10} + 95600065862285 T^{12} - 1134094607338 p^{2} T^{14} + 10014485938 p^{4} T^{16} - 64214018 p^{6} T^{18} + 283945 p^{8} T^{20} - 778 p^{10} T^{22} + p^{12} T^{24} \) |
| 79 | \( ( 1 + 14 T + 261 T^{2} + 3314 T^{3} + 40314 T^{4} + 387070 T^{5} + 3931233 T^{6} + 387070 p T^{7} + 40314 p^{2} T^{8} + 3314 p^{3} T^{9} + 261 p^{4} T^{10} + 14 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 83 | \( ( 1 + 20 T + 356 T^{2} + 3156 T^{3} + 24185 T^{4} + 42500 T^{5} + 239906 T^{6} + 42500 p T^{7} + 24185 p^{2} T^{8} + 3156 p^{3} T^{9} + 356 p^{4} T^{10} + 20 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 89 | \( ( 1 + 20 T + 338 T^{2} + 4896 T^{3} + 65639 T^{4} + 706796 T^{5} + 6957434 T^{6} + 706796 p T^{7} + 65639 p^{2} T^{8} + 4896 p^{3} T^{9} + 338 p^{4} T^{10} + 20 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 97 | \( 1 - 1072 T^{2} + 535030 T^{4} - 164321096 T^{6} + 34577082223 T^{8} - 5250510233512 T^{10} + 589883979492356 T^{12} - 5250510233512 p^{2} T^{14} + 34577082223 p^{4} T^{16} - 164321096 p^{6} T^{18} + 535030 p^{8} T^{20} - 1072 p^{10} T^{22} + p^{12} T^{24} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.33739575669580314889070589710, −2.12349658218570762823112301340, −2.11339758511298202383221160718, −2.09075877452166890297541818124, −1.95521245572495414740297147122, −1.94503844544670169870148071365, −1.93896502656517594827359541844, −1.64658022955209335018379354247, −1.45121254565525005911078549570, −1.41426945538619057262345622419, −1.28458539700699069135887779864, −1.25141266647530949161847688497, −1.19801587135602101741384484147, −1.18112562879885257544446108760, −1.11200854711268177718269834076, −1.00436066189669112297027053113, −0.965188495946569563495258981509, −0.917523505030216052924157012771, −0.834997819906089008323018067760, −0.51988032205579300792582433018, −0.35712560369384037939295901877, −0.22771734098233641415276551736, −0.17832029650219279202390169015, −0.12545744425597902959863978740, −0.05610106808497856506895067212,
0.05610106808497856506895067212, 0.12545744425597902959863978740, 0.17832029650219279202390169015, 0.22771734098233641415276551736, 0.35712560369384037939295901877, 0.51988032205579300792582433018, 0.834997819906089008323018067760, 0.917523505030216052924157012771, 0.965188495946569563495258981509, 1.00436066189669112297027053113, 1.11200854711268177718269834076, 1.18112562879885257544446108760, 1.19801587135602101741384484147, 1.25141266647530949161847688497, 1.28458539700699069135887779864, 1.41426945538619057262345622419, 1.45121254565525005911078549570, 1.64658022955209335018379354247, 1.93896502656517594827359541844, 1.94503844544670169870148071365, 1.95521245572495414740297147122, 2.09075877452166890297541818124, 2.11339758511298202383221160718, 2.12349658218570762823112301340, 2.33739575669580314889070589710
Plot not available for L-functions of degree greater than 10.