| L(s) = 1 | + 2-s + 4-s − 2·5-s + 8-s + 9-s − 2·10-s − 4·13-s + 16-s − 3·17-s + 18-s − 2·20-s + 3·25-s − 4·26-s + 32-s − 3·34-s + 36-s − 5·37-s − 2·40-s + 18·41-s − 2·45-s − 4·49-s + 3·50-s − 4·52-s − 15·53-s − 2·61-s + 64-s + 8·65-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 − 1.10·13-s + 1/4·16-s − 0.727·17-s + 0.235·18-s − 0.447·20-s + 3/5·25-s − 0.784·26-s + 0.176·32-s − 0.514·34-s + 1/6·36-s − 0.821·37-s − 0.316·40-s + 2.81·41-s − 0.298·45-s − 4/7·49-s + 0.424·50-s − 0.554·52-s − 2.06·53-s − 0.256·61-s + 1/8·64-s + 0.992·65-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 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 76 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 59 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−7.72121985228677483862036829683, −7.49517796274105006837950951789, −6.86213161922873404935799612575, −6.53705945148899152763716647420, −6.07409446571411463654609246311, −5.53093069630021108320006510524, −4.89321850998165364544794753340, −4.59916626397220222374706767589, −4.32300224604314100942428389937, −3.58248005966021909260734263226, −3.27062401346983310801314432572, −2.49070924863559179578543537232, −2.11517487993301540810123250676, −1.11684169631751583635508405373, 0,
1.11684169631751583635508405373, 2.11517487993301540810123250676, 2.49070924863559179578543537232, 3.27062401346983310801314432572, 3.58248005966021909260734263226, 4.32300224604314100942428389937, 4.59916626397220222374706767589, 4.89321850998165364544794753340, 5.53093069630021108320006510524, 6.07409446571411463654609246311, 6.53705945148899152763716647420, 6.86213161922873404935799612575, 7.49517796274105006837950951789, 7.72121985228677483862036829683
