| L(s) = 1 | + 2-s + 4-s − 2·5-s + 8-s + 9-s − 2·10-s + 2·13-s + 16-s + 18-s − 2·20-s + 3·25-s + 2·26-s + 32-s + 36-s + 4·37-s − 2·40-s − 12·41-s − 2·45-s − 10·49-s + 3·50-s + 2·52-s − 12·53-s − 20·61-s + 64-s − 4·65-s + 72-s + 16·73-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.353·8-s + 1/3·9-s − 0.632·10-s + 0.554·13-s + 1/4·16-s + 0.235·18-s − 0.447·20-s + 3/5·25-s + 0.392·26-s + 0.176·32-s + 1/6·36-s + 0.657·37-s − 0.316·40-s − 1.87·41-s − 0.298·45-s − 1.42·49-s + 0.424·50-s + 0.277·52-s − 1.64·53-s − 2.56·61-s + 1/8·64-s − 0.496·65-s + 0.117·72-s + 1.87·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−7.69992577245459400562531294135, −7.49770798161237830586232709263, −6.66017178643695125398418698543, −6.56327661741499604036243526620, −6.22443991483197983398572531638, −5.30751325359973115442660542578, −5.21150267903997949615096900503, −4.47211016896422697805960266468, −4.27346273235141665749062775801, −3.55157813952013317873025505155, −3.28257229246751367368737648325, −2.71877712145852755584836718019, −1.81386816189337101345457960761, −1.26481033481379490689403120538, 0,
1.26481033481379490689403120538, 1.81386816189337101345457960761, 2.71877712145852755584836718019, 3.28257229246751367368737648325, 3.55157813952013317873025505155, 4.27346273235141665749062775801, 4.47211016896422697805960266468, 5.21150267903997949615096900503, 5.30751325359973115442660542578, 6.22443991483197983398572531638, 6.56327661741499604036243526620, 6.66017178643695125398418698543, 7.49770798161237830586232709263, 7.69992577245459400562531294135
