| L(s) = 1 | + 2·4-s − 5-s − 8·7-s − 6·11-s − 2·13-s + 6·17-s − 7·19-s − 2·20-s − 16·28-s − 3·29-s − 14·31-s + 8·35-s + 16·37-s − 6·41-s + 4·43-s − 12·44-s + 6·47-s + 34·49-s − 4·52-s − 6·53-s + 6·55-s − 15·59-s − 5·61-s − 8·64-s + 2·65-s − 2·67-s + 12·68-s + ⋯ |
| L(s) = 1 | + 4-s − 0.447·5-s − 3.02·7-s − 1.80·11-s − 0.554·13-s + 1.45·17-s − 1.60·19-s − 0.447·20-s − 3.02·28-s − 0.557·29-s − 2.51·31-s + 1.35·35-s + 2.63·37-s − 0.937·41-s + 0.609·43-s − 1.80·44-s + 0.875·47-s + 34/7·49-s − 0.554·52-s − 0.824·53-s + 0.809·55-s − 1.95·59-s − 0.640·61-s − 64-s + 0.248·65-s − 0.244·67-s + 1.45·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 19 | $C_2$ | \( 1 + 7 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 15 T + 166 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 3 T - 62 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 8 T - 9 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 5 T - 54 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 15 T + 136 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 8 T - 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−9.850530987711527256674952408026, −9.785501073806385524056213412322, −9.072292990423787685858194721242, −9.037732842181625587261620891294, −7.899332607614189981339704991996, −7.88682278198065847964142894844, −7.32401323469590469361745876697, −6.97271323239509172320534047471, −6.55010406227323394050438693908, −6.07952880579286187774227883498, −5.58331007864135834733462007136, −5.51980529134237890805706407274, −4.18387938804896131922298628918, −4.17893331521981998392333728543, −3.05477246548069317452286738383, −3.01920831431247113420676509838, −2.66935415676807917451504337439, −1.77095464773124816393736803881, 0, 0,
1.77095464773124816393736803881, 2.66935415676807917451504337439, 3.01920831431247113420676509838, 3.05477246548069317452286738383, 4.17893331521981998392333728543, 4.18387938804896131922298628918, 5.51980529134237890805706407274, 5.58331007864135834733462007136, 6.07952880579286187774227883498, 6.55010406227323394050438693908, 6.97271323239509172320534047471, 7.32401323469590469361745876697, 7.88682278198065847964142894844, 7.899332607614189981339704991996, 9.037732842181625587261620891294, 9.072292990423787685858194721242, 9.785501073806385524056213412322, 9.850530987711527256674952408026
