| 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
| \(L(1)\) |
\(\approx\) |
\(2271.007110\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2271.007110\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1072 T^{2} + 535030 T^{4} - 164321096 T^{6} + 34577082223 T^{8} - 5250510233512 T^{10} + 589883979492356 T^{12} - 5250510233512 p^{2} T^{14} + 34577082223 p^{4} T^{16} - 164321096 p^{6} T^{18} + 535030 p^{8} T^{20} - 1072 p^{10} T^{22} + p^{12} T^{24} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.33995398763549682605526008876, −2.11812528035640403084340566926, −2.02002318770653682498482488086, −1.93664461742333909931296362952, −1.92766250717594621733930181496, −1.86814205128254046374971502097, −1.80996456779284878386072526913, −1.77517028939162620954460757366, −1.73010178861723664332880914820, −1.70374682264125157661032519265, −1.58285896488390651647377925877, −1.54341827445878254896276243763, −1.27712802784988966561841954575, −1.14711968842315383609180436946, −1.14496247899404786932055562759, −1.08769706850272936544623876378, −0.811433786014273845463716638873, −0.795034987475143875455515168744, −0.70750556693499217078324332374, −0.69955983922053859754253556337, −0.62824449992171550882869052101, −0.61757978857984158531981627638, −0.53931011777363910145763057705, −0.38513551597243584590912305500, −0.32991635147575727100624257857,
0.32991635147575727100624257857, 0.38513551597243584590912305500, 0.53931011777363910145763057705, 0.61757978857984158531981627638, 0.62824449992171550882869052101, 0.69955983922053859754253556337, 0.70750556693499217078324332374, 0.795034987475143875455515168744, 0.811433786014273845463716638873, 1.08769706850272936544623876378, 1.14496247899404786932055562759, 1.14711968842315383609180436946, 1.27712802784988966561841954575, 1.54341827445878254896276243763, 1.58285896488390651647377925877, 1.70374682264125157661032519265, 1.73010178861723664332880914820, 1.77517028939162620954460757366, 1.80996456779284878386072526913, 1.86814205128254046374971502097, 1.92766250717594621733930181496, 1.93664461742333909931296362952, 2.02002318770653682498482488086, 2.11812528035640403084340566926, 2.33995398763549682605526008876
Plot not available for L-functions of degree greater than 10.
