| L(s) = 1 | − 2-s + 2.68·3-s + 4-s + 0.637·5-s − 2.68·6-s − 2.14·7-s − 8-s + 4.20·9-s − 0.637·10-s + 4.14·11-s + 2.68·12-s + 13-s + 2.14·14-s + 1.71·15-s + 16-s − 4.09·17-s − 4.20·18-s + 0.637·20-s − 5.75·21-s − 4.14·22-s − 5.41·23-s − 2.68·24-s − 4.59·25-s − 26-s + 3.23·27-s − 2.14·28-s − 0.983·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.54·3-s + 0.5·4-s + 0.285·5-s − 1.09·6-s − 0.811·7-s − 0.353·8-s + 1.40·9-s − 0.201·10-s + 1.24·11-s + 0.774·12-s + 0.277·13-s + 0.573·14-s + 0.442·15-s + 0.250·16-s − 0.992·17-s − 0.990·18-s + 0.142·20-s − 1.25·21-s − 0.882·22-s − 1.12·23-s − 0.547·24-s − 0.918·25-s − 0.196·26-s + 0.621·27-s − 0.405·28-s − 0.182·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 2.68T + 3T^{2} \) |
| 5 | \( 1 - 0.637T + 5T^{2} \) |
| 7 | \( 1 + 2.14T + 7T^{2} \) |
| 11 | \( 1 - 4.14T + 11T^{2} \) |
| 17 | \( 1 + 4.09T + 17T^{2} \) |
| 23 | \( 1 + 5.41T + 23T^{2} \) |
| 29 | \( 1 + 0.983T + 29T^{2} \) |
| 31 | \( 1 + 6.46T + 31T^{2} \) |
| 37 | \( 1 + 1.72T + 37T^{2} \) |
| 41 | \( 1 + 9.07T + 41T^{2} \) |
| 43 | \( 1 - 0.899T + 43T^{2} \) |
| 47 | \( 1 + 5.20T + 47T^{2} \) |
| 53 | \( 1 - 5.58T + 53T^{2} \) |
| 59 | \( 1 + 4.94T + 59T^{2} \) |
| 61 | \( 1 - 5.38T + 61T^{2} \) |
| 67 | \( 1 + 2.12T + 67T^{2} \) |
| 71 | \( 1 - 8.02T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 + 13.1T + 83T^{2} \) |
| 89 | \( 1 - 13.4T + 89T^{2} \) |
| 97 | \( 1 + 17.0T + 97T^{2} \) |
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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
−7.52941266915103360247431895777, −6.73226496590409754750014279830, −6.40550282183643638475056345937, −5.44720334133142743304884451576, −4.06526639771770645152622560287, −3.75302372381614266726677384080, −2.95298441905368305959810260426, −2.03732925310576392885868651813, −1.58902659676538266994323546686, 0,
1.58902659676538266994323546686, 2.03732925310576392885868651813, 2.95298441905368305959810260426, 3.75302372381614266726677384080, 4.06526639771770645152622560287, 5.44720334133142743304884451576, 6.40550282183643638475056345937, 6.73226496590409754750014279830, 7.52941266915103360247431895777
