| L(s) = 1 | + 2-s + 0.594·3-s + 4-s − 1.59·5-s + 0.594·6-s − 7-s + 8-s − 2.64·9-s − 1.59·10-s − 5.13·11-s + 0.594·12-s + 2.64·13-s − 14-s − 0.947·15-s + 16-s + 6.54·17-s − 2.64·18-s + 5.73·19-s − 1.59·20-s − 0.594·21-s − 5.13·22-s + 0.594·24-s − 2.45·25-s + 2.64·26-s − 3.35·27-s − 28-s + 7.54·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.343·3-s + 0.5·4-s − 0.712·5-s + 0.242·6-s − 0.377·7-s + 0.353·8-s − 0.882·9-s − 0.504·10-s − 1.54·11-s + 0.171·12-s + 0.734·13-s − 0.267·14-s − 0.244·15-s + 0.250·16-s + 1.58·17-s − 0.623·18-s + 1.31·19-s − 0.356·20-s − 0.129·21-s − 1.09·22-s + 0.121·24-s − 0.491·25-s + 0.519·26-s − 0.645·27-s − 0.188·28-s + 1.40·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 0.594T + 3T^{2} \) |
| 5 | \( 1 + 1.59T + 5T^{2} \) |
| 11 | \( 1 + 5.13T + 11T^{2} \) |
| 13 | \( 1 - 2.64T + 13T^{2} \) |
| 17 | \( 1 - 6.54T + 17T^{2} \) |
| 19 | \( 1 - 5.73T + 19T^{2} \) |
| 29 | \( 1 - 7.54T + 29T^{2} \) |
| 31 | \( 1 - 4.24T + 31T^{2} \) |
| 37 | \( 1 + 6.24T + 37T^{2} \) |
| 41 | \( 1 + 5.83T + 41T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 + 2.08T + 47T^{2} \) |
| 53 | \( 1 + 9.54T + 53T^{2} \) |
| 59 | \( 1 - 8.78T + 59T^{2} \) |
| 61 | \( 1 + 13.5T + 61T^{2} \) |
| 67 | \( 1 + 1.49T + 67T^{2} \) |
| 71 | \( 1 + 1.67T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 + 9.42T + 79T^{2} \) |
| 83 | \( 1 + 7.64T + 83T^{2} \) |
| 89 | \( 1 - 1.67T + 89T^{2} \) |
| 97 | \( 1 + 6.80T + 97T^{2} \) |
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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
−7.72895468529934778018875261227, −6.84024725013128627514983618547, −6.00019735222057738208681126329, −5.33682789043002034022577949282, −4.87833991152849174804939255443, −3.64891240876972491041663496855, −3.19333758495874698209206759543, −2.74227088707360559890765159381, −1.36513365509348545548053913345, 0,
1.36513365509348545548053913345, 2.74227088707360559890765159381, 3.19333758495874698209206759543, 3.64891240876972491041663496855, 4.87833991152849174804939255443, 5.33682789043002034022577949282, 6.00019735222057738208681126329, 6.84024725013128627514983618547, 7.72895468529934778018875261227
