| L(s) = 1 | + 120·29-s − 144·41-s − 190·49-s + 140·61-s + 576·89-s + 192·101-s − 676·109-s + 460·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 434·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | + 4.13·29-s − 3.51·41-s − 3.87·49-s + 2.29·61-s + 6.47·89-s + 1.90·101-s − 6.20·109-s + 3.80·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 2.56·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(8.215295434\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.215295434\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( ( 1 + 95 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 230 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 217 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 542 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 359 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 758 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 30 T + p^{2} T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 1895 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 2542 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 36 T + p^{2} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 3671 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 2390 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 434 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 5510 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 35 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 8111 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 1970 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 6814 T^{2} + p^{4} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 6674 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 8486 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 144 T + p^{2} T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 + 13943 T^{2} + p^{4} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.98601871565260168336057595173, −5.60123611798990454638520754203, −5.43833031998891729078997015437, −5.14689785992628541075918217673, −5.04222035586740224346478913851, −4.83300534300147186419310915829, −4.61733122849422531587435199175, −4.55756000215471912977094271916, −4.54585408276919555276118786222, −3.89994893035089462330515437448, −3.70283520851719500716941461187, −3.64901579931128479106789092060, −3.38372123455330715816123673627, −3.07764478949507253547930168054, −2.99683446162355510567446935059, −2.67332048813011186453698147214, −2.65808468752695164347760794912, −2.00020875711299587243868291629, −1.90040069374203479891061898841, −1.78783505625559019139549907199, −1.51939231160244852644030466314, −1.03746690498481341029406480971, −0.75230876676792442455312679868, −0.49244938311742954311558013436, −0.40200268374529897930758281040,
0.40200268374529897930758281040, 0.49244938311742954311558013436, 0.75230876676792442455312679868, 1.03746690498481341029406480971, 1.51939231160244852644030466314, 1.78783505625559019139549907199, 1.90040069374203479891061898841, 2.00020875711299587243868291629, 2.65808468752695164347760794912, 2.67332048813011186453698147214, 2.99683446162355510567446935059, 3.07764478949507253547930168054, 3.38372123455330715816123673627, 3.64901579931128479106789092060, 3.70283520851719500716941461187, 3.89994893035089462330515437448, 4.54585408276919555276118786222, 4.55756000215471912977094271916, 4.61733122849422531587435199175, 4.83300534300147186419310915829, 5.04222035586740224346478913851, 5.14689785992628541075918217673, 5.43833031998891729078997015437, 5.60123611798990454638520754203, 5.98601871565260168336057595173
