| L(s) = 1 | − 2-s + 4-s + 2·5-s − 8-s + 9-s − 2·10-s + 16-s − 2·17-s − 18-s + 2·20-s − 25-s − 2·29-s − 32-s + 2·34-s + 36-s + 4·37-s − 2·40-s + 8·41-s + 2·45-s − 6·49-s + 50-s − 4·53-s + 2·58-s − 28·61-s + 64-s − 2·68-s − 72-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 + 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.447·20-s − 1/5·25-s − 0.371·29-s − 0.176·32-s + 0.342·34-s + 1/6·36-s + 0.657·37-s − 0.316·40-s + 1.24·41-s + 0.298·45-s − 6/7·49-s + 0.141·50-s − 0.549·53-s + 0.262·58-s − 3.58·61-s + 1/8·64-s − 0.242·68-s − 0.117·72-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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 12 T + p T^{2} ) \) |
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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
−7.81243856015188900327511512091, −7.56165374565566633434351609153, −6.85522800548892822248990463772, −6.54976459248081974492991425482, −6.09325397707700255516488439606, −5.74078842980022542466432894157, −5.27154670353916210592319398356, −4.48745290394150712677109023825, −4.34430054843598245496948623591, −3.47933304681142746672259344341, −2.91314926964116352374060722366, −2.37044623892429600211586105475, −1.73706108805119776626811752891, −1.23013537793394780561967084251, 0,
1.23013537793394780561967084251, 1.73706108805119776626811752891, 2.37044623892429600211586105475, 2.91314926964116352374060722366, 3.47933304681142746672259344341, 4.34430054843598245496948623591, 4.48745290394150712677109023825, 5.27154670353916210592319398356, 5.74078842980022542466432894157, 6.09325397707700255516488439606, 6.54976459248081974492991425482, 6.85522800548892822248990463772, 7.56165374565566633434351609153, 7.81243856015188900327511512091
