| L(s) = 1 | + 4·8-s + 4·13-s + 3·16-s + 20·17-s − 4·32-s − 4·37-s + 16·41-s − 4·53-s − 32·61-s + 8·64-s − 44·73-s − 3·81-s + 20·97-s − 40·101-s + 16·104-s + 52·113-s + 60·121-s + 127-s + 16·128-s + 131-s + 80·136-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 1.41·8-s + 1.10·13-s + 3/4·16-s + 4.85·17-s − 0.707·32-s − 0.657·37-s + 2.49·41-s − 0.549·53-s − 4.09·61-s + 64-s − 5.14·73-s − 1/3·81-s + 2.03·97-s − 3.98·101-s + 1.56·104-s + 4.89·113-s + 5.45·121-s + 0.0887·127-s + 1.41·128-s + 0.0873·131-s + 6.85·136-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.357976851\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.357976851\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T^{3} - 3 T^{4} + p^{2} T^{5} + p^{3} T^{6} + p^{3} T^{7} - 3 p^{2} T^{8} - p^{5} T^{9} + p^{6} T^{12} \) |
| 3 | \( ( 1 + T^{4} )^{3} \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 66 T^{4} + 671 T^{8} + 41476 T^{12} + 671 p^{4} T^{16} - 66 p^{8} T^{20} + p^{12} T^{24} \) |
| 11 | \( ( 1 - 30 T^{2} + 491 T^{4} - 6076 T^{6} + 491 p^{2} T^{8} - 30 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 13 | \( ( 1 - 2 T + 2 T^{2} + 6 T^{3} + 27 T^{4} - 580 T^{5} + 1124 T^{6} - 580 p T^{7} + 27 p^{2} T^{8} + 6 p^{3} T^{9} + 2 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 17 | \( ( 1 - 10 T + 50 T^{2} - 250 T^{3} + 1155 T^{4} - 4260 T^{5} + 16100 T^{6} - 4260 p T^{7} + 1155 p^{2} T^{8} - 250 p^{3} T^{9} + 50 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 19 | \( ( 1 + 74 T^{2} + 2775 T^{4} + 65228 T^{6} + 2775 p^{2} T^{8} + 74 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 23 | \( 1 + 1190 T^{4} + 607727 T^{8} + 223349076 T^{12} + 607727 p^{4} T^{16} + 1190 p^{8} T^{20} + p^{12} T^{24} \) |
| 29 | \( ( 1 - 154 T^{2} + 10395 T^{4} - 392596 T^{6} + 10395 p^{2} T^{8} - 154 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 31 | \( ( 1 - 74 T^{2} + 4175 T^{4} - 143436 T^{6} + 4175 p^{2} T^{8} - 74 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 37 | \( ( 1 - 8 T - 11 T^{2} + 424 T^{3} - 11 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2}( 1 + 10 T + 93 T^{2} + 472 T^{3} + 93 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 41 | \( ( 1 - 4 T + 103 T^{2} - 264 T^{3} + 103 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 43 | \( 1 - 6826 T^{4} + 23542879 T^{8} - 52748850124 T^{12} + 23542879 p^{4} T^{16} - 6826 p^{8} T^{20} + p^{12} T^{24} \) |
| 47 | \( 1 + 1606 T^{4} + 1555663 T^{8} - 11534742380 T^{12} + 1555663 p^{4} T^{16} + 1606 p^{8} T^{20} + p^{12} T^{24} \) |
| 53 | \( ( 1 + 2 T + 2 T^{2} + 90 T^{3} + 5291 T^{4} + 7252 T^{5} + 7972 T^{6} + 7252 p T^{7} + 5291 p^{2} T^{8} + 90 p^{3} T^{9} + 2 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 59 | \( ( 1 + 254 T^{2} + 30475 T^{4} + 2237948 T^{6} + 30475 p^{2} T^{8} + 254 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 61 | \( ( 1 + 8 T + 83 T^{2} + 1152 T^{3} + 83 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 67 | \( 1 - 6346 T^{4} + 60212671 T^{8} - 234156159244 T^{12} + 60212671 p^{4} T^{16} - 6346 p^{8} T^{20} + p^{12} T^{24} \) |
| 71 | \( ( 1 - 170 T^{2} + 18527 T^{4} - 1427916 T^{6} + 18527 p^{2} T^{8} - 170 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 73 | \( ( 1 + 22 T + 242 T^{2} + 2246 T^{3} + 20271 T^{4} + 179604 T^{5} + 1567964 T^{6} + 179604 p T^{7} + 20271 p^{2} T^{8} + 2246 p^{3} T^{9} + 242 p^{4} T^{10} + 22 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
| 79 | \( ( 1 + 170 T^{2} + 10415 T^{4} + 507660 T^{6} + 10415 p^{2} T^{8} + 170 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 83 | \( 1 + 7542 T^{4} - 16589569 T^{8} - 295904039564 T^{12} - 16589569 p^{4} T^{16} + 7542 p^{8} T^{20} + p^{12} T^{24} \) |
| 89 | \( ( 1 - 462 T^{2} + 94223 T^{4} - 10861604 T^{6} + 94223 p^{2} T^{8} - 462 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 97 | \( ( 1 - 10 T + 50 T^{2} + 1078 T^{3} - 9153 T^{4} - 45804 T^{5} + 1496732 T^{6} - 45804 p T^{7} - 9153 p^{2} T^{8} + 1078 p^{3} T^{9} + 50 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} )^{2} \) |
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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
−4.06999611731451464439701201278, −4.03987637093437876168409431052, −3.63456712892434686927069440608, −3.57342429284896941660855445346, −3.50648444494165044487245167749, −3.46971142264120897335295565981, −3.41693746278394688542196685161, −3.19485994822676857198822705381, −3.13535207034008008661760829833, −3.05331725450058106902687451682, −2.93765893526049545430800879575, −2.78363854866929455152207567159, −2.74274511803492942063864279541, −2.50397301425774830549780813618, −2.33232371828655254728883350738, −2.04101588244523753159519780424, −1.86933103756159213637693505089, −1.84935072363731228825814436478, −1.58987505401084893253416031023, −1.57815415673670337045910947944, −1.29473534620360346379554490425, −1.22331731144603691286806814467, −1.02625442251022805317732936383, −0.866095057040005685096438766995, −0.44836627200866027856961988057,
0.44836627200866027856961988057, 0.866095057040005685096438766995, 1.02625442251022805317732936383, 1.22331731144603691286806814467, 1.29473534620360346379554490425, 1.57815415673670337045910947944, 1.58987505401084893253416031023, 1.84935072363731228825814436478, 1.86933103756159213637693505089, 2.04101588244523753159519780424, 2.33232371828655254728883350738, 2.50397301425774830549780813618, 2.74274511803492942063864279541, 2.78363854866929455152207567159, 2.93765893526049545430800879575, 3.05331725450058106902687451682, 3.13535207034008008661760829833, 3.19485994822676857198822705381, 3.41693746278394688542196685161, 3.46971142264120897335295565981, 3.50648444494165044487245167749, 3.57342429284896941660855445346, 3.63456712892434686927069440608, 4.03987637093437876168409431052, 4.06999611731451464439701201278
Plot not available for L-functions of degree greater than 10.
