| L(s) = 1 | + 2·5-s + 7-s − 5·11-s − 2·13-s − 5·17-s + 4·19-s + 7·23-s + 3·25-s − 6·29-s + 6·31-s + 2·35-s − 7·37-s − 15·41-s − 8·43-s + 10·47-s − 3·49-s − 15·53-s − 10·55-s − 4·59-s − 61-s − 4·65-s + 8·67-s − 71-s + 14·73-s − 5·77-s − 3·79-s + 8·83-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.377·7-s − 1.50·11-s − 0.554·13-s − 1.21·17-s + 0.917·19-s + 1.45·23-s + 3/5·25-s − 1.11·29-s + 1.07·31-s + 0.338·35-s − 1.15·37-s − 2.34·41-s − 1.21·43-s + 1.45·47-s − 3/7·49-s − 2.06·53-s − 1.34·55-s − 0.520·59-s − 0.128·61-s − 0.496·65-s + 0.977·67-s − 0.118·71-s + 1.63·73-s − 0.569·77-s − 0.337·79-s + 0.878·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87609600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87609600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - T + 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 5 T + 18 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 5 T + 30 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 7 T + 48 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 7 T + 76 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 15 T + 128 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 10 T + 78 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 15 T + 152 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T - 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + T + 112 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + T - 114 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 14 T + 154 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 3 T + 150 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 3 T + 88 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 17 T + 174 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
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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
−7.43256422827922948991129983681, −7.22387541655726001651855176679, −6.80595202641707762156583189827, −6.62226144316176527223879867038, −6.17309326420401392085496277060, −5.83119054922370442737687881858, −5.19238154745604756079512964059, −5.12243531686133960522812409115, −4.86634794768885878280614298190, −4.81531237459341159737338127599, −3.96774355446736041076192547713, −3.48202183009771130119817478516, −3.25971696329102952579684189234, −2.72754347560408155194838571386, −2.27044806234581786474247966073, −2.20069346990799832332069687987, −1.36589485184589014686133084388, −1.23799142159083641139576355625, 0, 0,
1.23799142159083641139576355625, 1.36589485184589014686133084388, 2.20069346990799832332069687987, 2.27044806234581786474247966073, 2.72754347560408155194838571386, 3.25971696329102952579684189234, 3.48202183009771130119817478516, 3.96774355446736041076192547713, 4.81531237459341159737338127599, 4.86634794768885878280614298190, 5.12243531686133960522812409115, 5.19238154745604756079512964059, 5.83119054922370442737687881858, 6.17309326420401392085496277060, 6.62226144316176527223879867038, 6.80595202641707762156583189827, 7.22387541655726001651855176679, 7.43256422827922948991129983681
