| L(s) = 1 | − 2-s − 1.19·3-s + 4-s − 2.13·5-s + 1.19·6-s − 8-s − 1.56·9-s + 2.13·10-s − 4.56·11-s − 1.19·12-s + 4.53·13-s + 2.56·15-s + 16-s − 5.20·17-s + 1.56·18-s + 6.14·19-s − 2.13·20-s + 4.56·22-s + 8.24·23-s + 1.19·24-s − 0.438·25-s − 4.53·26-s + 5.47·27-s + 29-s − 2.56·30-s + 4.53·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.692·3-s + 0.5·4-s − 0.955·5-s + 0.489·6-s − 0.353·8-s − 0.520·9-s + 0.675·10-s − 1.37·11-s − 0.346·12-s + 1.25·13-s + 0.661·15-s + 0.250·16-s − 1.26·17-s + 0.368·18-s + 1.40·19-s − 0.477·20-s + 0.972·22-s + 1.71·23-s + 0.244·24-s − 0.0876·25-s − 0.889·26-s + 1.05·27-s + 0.185·29-s − 0.467·30-s + 0.814·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + 1.19T + 3T^{2} \) |
| 5 | \( 1 + 2.13T + 5T^{2} \) |
| 11 | \( 1 + 4.56T + 11T^{2} \) |
| 13 | \( 1 - 4.53T + 13T^{2} \) |
| 17 | \( 1 + 5.20T + 17T^{2} \) |
| 19 | \( 1 - 6.14T + 19T^{2} \) |
| 23 | \( 1 - 8.24T + 23T^{2} \) |
| 31 | \( 1 - 4.53T + 31T^{2} \) |
| 37 | \( 1 + 1.12T + 37T^{2} \) |
| 41 | \( 1 + 5.20T + 41T^{2} \) |
| 43 | \( 1 - 7.68T + 43T^{2} \) |
| 47 | \( 1 + 2.13T + 47T^{2} \) |
| 53 | \( 1 + 7.43T + 53T^{2} \) |
| 59 | \( 1 - 5.20T + 59T^{2} \) |
| 61 | \( 1 - 4.27T + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 + 13.3T + 71T^{2} \) |
| 73 | \( 1 - 16.6T + 73T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 - 10.0T + 83T^{2} \) |
| 89 | \( 1 + 2.80T + 89T^{2} \) |
| 97 | \( 1 + 13.7T + 97T^{2} \) |
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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
−8.460609029444875526923068800902, −7.72115276503180393417544508236, −7.02460006398578149325072025033, −6.19764330858997277098079741178, −5.37020872563565395947631442802, −4.63603661540254214533212858966, −3.38312255693820529430211035809, −2.67473567460688958214678111473, −1.06105388899107994111568830543, 0,
1.06105388899107994111568830543, 2.67473567460688958214678111473, 3.38312255693820529430211035809, 4.63603661540254214533212858966, 5.37020872563565395947631442802, 6.19764330858997277098079741178, 7.02460006398578149325072025033, 7.72115276503180393417544508236, 8.460609029444875526923068800902
