| L(s) = 1 | − 6·3-s + 16·4-s + 27·9-s − 96·12-s + 192·16-s − 234·17-s + 36·23-s + 223·25-s − 108·27-s − 198·29-s + 432·36-s − 164·43-s − 1.15e3·48-s + 578·49-s + 1.40e3·51-s − 522·53-s − 1.43e3·61-s + 2.04e3·64-s − 3.74e3·68-s − 216·69-s − 1.33e3·75-s − 880·79-s + 405·81-s + 1.18e3·87-s + 576·92-s + 3.56e3·100-s + 3.15e3·101-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 2·4-s + 9-s − 2.30·12-s + 3·16-s − 3.33·17-s + 0.326·23-s + 1.78·25-s − 0.769·27-s − 1.26·29-s + 2·36-s − 0.581·43-s − 3.46·48-s + 1.68·49-s + 3.85·51-s − 1.35·53-s − 3.01·61-s + 4·64-s − 6.67·68-s − 0.376·69-s − 2.05·75-s − 1.25·79-s + 5/9·81-s + 1.46·87-s + 0.652·92-s + 3.56·100-s + 3.10·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.103060031\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.103060031\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 223 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 578 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 38 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 117 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 13130 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 18 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 99 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 21950 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 88631 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 136519 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 82 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 202354 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 261 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 213050 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 719 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 107018 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 497122 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 309959 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 440 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 284726 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 892190 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 486674 T^{2} + p^{6} T^{4} \) |
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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
−10.68146975157105867078297431534, −10.59435348400183086916922527739, −10.30434883002948537473483227863, −9.232427400492467480500414024920, −9.110307475348683644266951629779, −8.523804449304950013507301119182, −7.69352686227346422618522077820, −7.39201319710844994345600880938, −6.89568623477962945494789150529, −6.53130323875136454742302039664, −6.33221169210607497419078390249, −5.79927056157897466393676768853, −5.13397355008655610771556281911, −4.63445663751805936925757277499, −4.07515370850593451866546208597, −3.16696449929894139800470380346, −2.61907955212781802327804428360, −1.95192339247304154875575208710, −1.49165933183868599628163184016, −0.45477283372033126823421343651,
0.45477283372033126823421343651, 1.49165933183868599628163184016, 1.95192339247304154875575208710, 2.61907955212781802327804428360, 3.16696449929894139800470380346, 4.07515370850593451866546208597, 4.63445663751805936925757277499, 5.13397355008655610771556281911, 5.79927056157897466393676768853, 6.33221169210607497419078390249, 6.53130323875136454742302039664, 6.89568623477962945494789150529, 7.39201319710844994345600880938, 7.69352686227346422618522077820, 8.523804449304950013507301119182, 9.110307475348683644266951629779, 9.232427400492467480500414024920, 10.30434883002948537473483227863, 10.59435348400183086916922527739, 10.68146975157105867078297431534
