| L(s) = 1 | + 5·3-s − 7·5-s − 14·7-s − 9-s + 22·11-s + 46·13-s − 35·15-s − 88·17-s + 46·19-s − 70·21-s + 115·23-s − 179·25-s − 372·29-s + 137·31-s + 110·33-s + 98·35-s − 19·37-s + 230·39-s − 440·41-s − 192·43-s + 7·45-s − 728·47-s + 147·49-s − 440·51-s − 808·53-s − 154·55-s + 230·57-s + ⋯ |
| L(s) = 1 | + 0.962·3-s − 0.626·5-s − 0.755·7-s − 0.0370·9-s + 0.603·11-s + 0.981·13-s − 0.602·15-s − 1.25·17-s + 0.555·19-s − 0.727·21-s + 1.04·23-s − 1.43·25-s − 2.38·29-s + 0.793·31-s + 0.580·33-s + 0.473·35-s − 0.0844·37-s + 0.944·39-s − 1.67·41-s − 0.680·43-s + 0.0231·45-s − 2.25·47-s + 3/7·49-s − 1.20·51-s − 2.09·53-s − 0.377·55-s + 0.534·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1517824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1517824 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9329851605\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9329851605\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 5 T + 26 T^{2} - 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 7 T + 228 T^{2} + 7 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 46 T + 1498 T^{2} - 46 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 88 T + 11214 T^{2} + 88 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 46 T + 7534 T^{2} - 46 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 5 p T + 19934 T^{2} - 5 p^{4} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 372 T + 74606 T^{2} + 372 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 137 T + 63966 T^{2} - 137 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 19 T + 100540 T^{2} + 19 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 440 T + 172542 T^{2} + 440 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 192 T + 166038 T^{2} + 192 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 728 T + 320414 T^{2} + 728 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 808 T + 447270 T^{2} + 808 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 35 T + 238410 T^{2} - 35 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 312 T + 135798 T^{2} - 312 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 301 T - 177582 T^{2} - 301 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 137 T + 185358 T^{2} - 137 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 788 T + 714070 T^{2} - 788 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 366 T + 1018334 T^{2} - 366 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 28 p T + 2480118 T^{2} - 28 p^{4} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 1235 T + 1776140 T^{2} - 1235 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 709 T + 828952 T^{2} - 709 p^{3} T^{3} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.432944170318231861550337333848, −9.303709212423638394413241788135, −8.522901896881293512684781113502, −8.502495432489321904088256663105, −7.979548423162149277999147856764, −7.67098828971359044050986776727, −6.98035435810834161250740411447, −6.77992282774079127536142026125, −6.17999645680566815557910420325, −6.07657734248593440532776979250, −5.07204832740554947810576911889, −4.96807116659622808166037014867, −4.20730622993960559534615824799, −3.53081233255917207235721551120, −3.43658912563391294004548104317, −3.24221831011902974824518140016, −2.10498870193823458937700646139, −1.99498523288508572864207090016, −1.10681016494464160829783328810, −0.22200957372730158737587154282,
0.22200957372730158737587154282, 1.10681016494464160829783328810, 1.99498523288508572864207090016, 2.10498870193823458937700646139, 3.24221831011902974824518140016, 3.43658912563391294004548104317, 3.53081233255917207235721551120, 4.20730622993960559534615824799, 4.96807116659622808166037014867, 5.07204832740554947810576911889, 6.07657734248593440532776979250, 6.17999645680566815557910420325, 6.77992282774079127536142026125, 6.98035435810834161250740411447, 7.67098828971359044050986776727, 7.979548423162149277999147856764, 8.502495432489321904088256663105, 8.522901896881293512684781113502, 9.303709212423638394413241788135, 9.432944170318231861550337333848
