| L(s) = 1 | + 4·5-s − 2·7-s − 9-s − 6·13-s + 11·25-s − 8·29-s − 8·35-s + 22·37-s − 4·45-s − 24·47-s − 11·49-s − 14·61-s + 2·63-s − 24·65-s − 8·67-s + 4·73-s + 30·79-s + 81-s − 24·83-s + 12·91-s − 34·97-s + 24·101-s + 6·117-s + 13·121-s + 24·125-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 0.755·7-s − 1/3·9-s − 1.66·13-s + 11/5·25-s − 1.48·29-s − 1.35·35-s + 3.61·37-s − 0.596·45-s − 3.50·47-s − 1.57·49-s − 1.79·61-s + 0.251·63-s − 2.97·65-s − 0.977·67-s + 0.468·73-s + 3.37·79-s + 1/9·81-s − 2.63·83-s + 1.25·91-s − 3.45·97-s + 2.38·101-s + 0.554·117-s + 1.18·121-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.566135408\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.566135408\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 61 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 153 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 17 T + p T^{2} )^{2} \) |
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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
−9.145435740646397734700566089888, −8.561948052798007621341784843487, −7.918342833191306543358269025067, −7.918164034877607257709138192859, −7.37583168496903417676030858258, −6.90835927493253780310569462498, −6.39007267168883579936805089241, −6.31479167954352004943190717887, −5.95621054970708431915266751796, −5.48886729243864627933331908888, −5.03753098480552359382494038470, −4.77160474010311017486387231983, −4.37871668056644707231104679370, −3.65253026553392531665876688659, −2.96075006783158182729391736254, −2.90899046425487248062137332998, −2.35771204582972671122281633947, −1.81105444955004457262087101811, −1.40758117694401192774726222538, −0.36591935044072881128055277779,
0.36591935044072881128055277779, 1.40758117694401192774726222538, 1.81105444955004457262087101811, 2.35771204582972671122281633947, 2.90899046425487248062137332998, 2.96075006783158182729391736254, 3.65253026553392531665876688659, 4.37871668056644707231104679370, 4.77160474010311017486387231983, 5.03753098480552359382494038470, 5.48886729243864627933331908888, 5.95621054970708431915266751796, 6.31479167954352004943190717887, 6.39007267168883579936805089241, 6.90835927493253780310569462498, 7.37583168496903417676030858258, 7.918164034877607257709138192859, 7.918342833191306543358269025067, 8.561948052798007621341784843487, 9.145435740646397734700566089888
