| 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 + 28·61-s + 64-s + 4·65-s − 72-s − 8·73-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 + 3.58·61-s + 1/8·64-s + 0.496·65-s − 0.117·72-s − 0.936·73-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
| \(L(1)\) |
\(\approx\) |
\(1.956895169\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.956895169\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
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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
−7.923933777636678991733597967480, −7.85832220713197698190585116152, −6.99686575372866242821342185369, −6.77145317316863409022172537659, −6.42262701034001882837697519636, −5.70233605244569122682417518238, −5.65253747259908769204779494812, −4.98674800015260782294546923298, −4.28133784539552676878054432798, −3.98803889555059043901076634706, −3.09638465664731709775430870739, −2.75082196840239005573414536478, −2.00888213860569869943119252787, −1.49299392430960775577654396970, −0.73199284761332162352741133507,
0.73199284761332162352741133507, 1.49299392430960775577654396970, 2.00888213860569869943119252787, 2.75082196840239005573414536478, 3.09638465664731709775430870739, 3.98803889555059043901076634706, 4.28133784539552676878054432798, 4.98674800015260782294546923298, 5.65253747259908769204779494812, 5.70233605244569122682417518238, 6.42262701034001882837697519636, 6.77145317316863409022172537659, 6.99686575372866242821342185369, 7.85832220713197698190585116152, 7.923933777636678991733597967480
