| L(s) = 1 | + 2.11·3-s + 2.26·5-s − 4.06·7-s + 1.46·9-s − 1.00·11-s − 5.09·13-s + 4.78·15-s + 1.45·17-s + 6.53·19-s − 8.58·21-s − 4.54·23-s + 0.136·25-s − 3.25·27-s − 6.23·29-s − 10.1·31-s − 2.11·33-s − 9.21·35-s + 3.44·37-s − 10.7·39-s − 9.13·41-s + 2.78·43-s + 3.31·45-s + 6.12·47-s + 9.52·49-s + 3.07·51-s − 3.12·53-s − 2.27·55-s + ⋯ |
| L(s) = 1 | + 1.21·3-s + 1.01·5-s − 1.53·7-s + 0.486·9-s − 0.302·11-s − 1.41·13-s + 1.23·15-s + 0.352·17-s + 1.50·19-s − 1.87·21-s − 0.948·23-s + 0.0272·25-s − 0.625·27-s − 1.15·29-s − 1.81·31-s − 0.368·33-s − 1.55·35-s + 0.566·37-s − 1.72·39-s − 1.42·41-s + 0.424·43-s + 0.493·45-s + 0.892·47-s + 1.36·49-s + 0.430·51-s − 0.428·53-s − 0.306·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 - 2.11T + 3T^{2} \) |
| 5 | \( 1 - 2.26T + 5T^{2} \) |
| 7 | \( 1 + 4.06T + 7T^{2} \) |
| 11 | \( 1 + 1.00T + 11T^{2} \) |
| 13 | \( 1 + 5.09T + 13T^{2} \) |
| 17 | \( 1 - 1.45T + 17T^{2} \) |
| 19 | \( 1 - 6.53T + 19T^{2} \) |
| 23 | \( 1 + 4.54T + 23T^{2} \) |
| 29 | \( 1 + 6.23T + 29T^{2} \) |
| 31 | \( 1 + 10.1T + 31T^{2} \) |
| 37 | \( 1 - 3.44T + 37T^{2} \) |
| 41 | \( 1 + 9.13T + 41T^{2} \) |
| 43 | \( 1 - 2.78T + 43T^{2} \) |
| 47 | \( 1 - 6.12T + 47T^{2} \) |
| 53 | \( 1 + 3.12T + 53T^{2} \) |
| 59 | \( 1 - 2.23T + 59T^{2} \) |
| 61 | \( 1 + 13.6T + 61T^{2} \) |
| 67 | \( 1 - 14.1T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 - 0.606T + 73T^{2} \) |
| 79 | \( 1 + 14.1T + 79T^{2} \) |
| 83 | \( 1 + 6.43T + 83T^{2} \) |
| 89 | \( 1 - 16.6T + 89T^{2} \) |
| 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
−7.974715872147891444271954134956, −7.44874625600768947448052523393, −6.75614644707235637235891900985, −5.68823058583902979949832524818, −5.37878749771890497893632531069, −3.90532007649302927653140226563, −3.23898333039314676147857742042, −2.55748028211285875392698272191, −1.83011373671736023689161484887, 0,
1.83011373671736023689161484887, 2.55748028211285875392698272191, 3.23898333039314676147857742042, 3.90532007649302927653140226563, 5.37878749771890497893632531069, 5.68823058583902979949832524818, 6.75614644707235637235891900985, 7.44874625600768947448052523393, 7.974715872147891444271954134956
