| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2433600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2433600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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
−8.957396008295751097470592684353, −8.656303373274142701147551261804, −8.597231495125292446000657893283, −8.171334309890511030019834182007, −7.44608206151356399470591992666, −7.43769923044482681721573208872, −6.83301670908058732581986352629, −6.51240925284470699965139942082, −5.94713115849714151891680690526, −5.69953932270501505527754892035, −4.80123272785916480515430985142, −4.74718527872847732493153955039, −4.04089543044068293810390344081, −3.37175817006205386659209044867, −3.28559050388715384126239820984, −3.21687578540934487657188580429, −1.78317035145230073166303829379, −1.45816352441962070690744311768, 0, 0,
1.45816352441962070690744311768, 1.78317035145230073166303829379, 3.21687578540934487657188580429, 3.28559050388715384126239820984, 3.37175817006205386659209044867, 4.04089543044068293810390344081, 4.74718527872847732493153955039, 4.80123272785916480515430985142, 5.69953932270501505527754892035, 5.94713115849714151891680690526, 6.51240925284470699965139942082, 6.83301670908058732581986352629, 7.43769923044482681721573208872, 7.44608206151356399470591992666, 8.171334309890511030019834182007, 8.597231495125292446000657893283, 8.656303373274142701147551261804, 8.957396008295751097470592684353
