| L(s) = 1 | − 2-s − 2·3-s − 4-s + 2·6-s + 2·7-s + 3·8-s + 9-s + 2·11-s + 2·12-s − 2·13-s − 2·14-s − 16-s − 17-s − 18-s − 4·21-s − 2·22-s − 6·23-s − 6·24-s + 2·26-s + 4·27-s − 2·28-s − 6·29-s − 10·31-s − 5·32-s − 4·33-s + 34-s − 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 1/2·4-s + 0.816·6-s + 0.755·7-s + 1.06·8-s + 1/3·9-s + 0.603·11-s + 0.577·12-s − 0.554·13-s − 0.534·14-s − 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.872·21-s − 0.426·22-s − 1.25·23-s − 1.22·24-s + 0.392·26-s + 0.769·27-s − 0.377·28-s − 1.11·29-s − 1.79·31-s − 0.883·32-s − 0.696·33-s + 0.171·34-s − 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.80523599860593932803878715836, −9.837106295977008288474085621762, −9.004724365548119637924595418752, −8.015272075570334712392855961758, −7.11494978016373279342190063323, −5.84510932829609015266318935243, −5.00348909499756108069005690763, −4.01843802221452326726380097386, −1.68337813127645467731093868081, 0,
1.68337813127645467731093868081, 4.01843802221452326726380097386, 5.00348909499756108069005690763, 5.84510932829609015266318935243, 7.11494978016373279342190063323, 8.015272075570334712392855961758, 9.004724365548119637924595418752, 9.837106295977008288474085621762, 10.80523599860593932803878715836
