| L(s) = 1 | − 2.26·2-s + 2.16·3-s + 3.12·4-s − 5-s − 4.89·6-s + 7-s − 2.54·8-s + 1.67·9-s + 2.26·10-s + 3.90·11-s + 6.74·12-s − 1.97·13-s − 2.26·14-s − 2.16·15-s − 0.493·16-s − 4.28·17-s − 3.78·18-s + 3.00·19-s − 3.12·20-s + 2.16·21-s − 8.83·22-s − 6.89·23-s − 5.49·24-s + 25-s + 4.47·26-s − 2.87·27-s + 3.12·28-s + ⋯ |
| L(s) = 1 | − 1.60·2-s + 1.24·3-s + 1.56·4-s − 0.447·5-s − 1.99·6-s + 0.377·7-s − 0.898·8-s + 0.556·9-s + 0.715·10-s + 1.17·11-s + 1.94·12-s − 0.548·13-s − 0.604·14-s − 0.558·15-s − 0.123·16-s − 1.04·17-s − 0.891·18-s + 0.690·19-s − 0.698·20-s + 0.471·21-s − 1.88·22-s − 1.43·23-s − 1.12·24-s + 0.200·25-s + 0.877·26-s − 0.552·27-s + 0.590·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 229 | \( 1 - T \) |
| good | 2 | \( 1 + 2.26T + 2T^{2} \) |
| 3 | \( 1 - 2.16T + 3T^{2} \) |
| 11 | \( 1 - 3.90T + 11T^{2} \) |
| 13 | \( 1 + 1.97T + 13T^{2} \) |
| 17 | \( 1 + 4.28T + 17T^{2} \) |
| 19 | \( 1 - 3.00T + 19T^{2} \) |
| 23 | \( 1 + 6.89T + 23T^{2} \) |
| 29 | \( 1 + 0.0129T + 29T^{2} \) |
| 31 | \( 1 - 2.71T + 31T^{2} \) |
| 37 | \( 1 + 6.17T + 37T^{2} \) |
| 41 | \( 1 + 3.02T + 41T^{2} \) |
| 43 | \( 1 + 4.37T + 43T^{2} \) |
| 47 | \( 1 - 12.7T + 47T^{2} \) |
| 53 | \( 1 - 3.81T + 53T^{2} \) |
| 59 | \( 1 - 10.9T + 59T^{2} \) |
| 61 | \( 1 - 0.472T + 61T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 - 2.95T + 71T^{2} \) |
| 73 | \( 1 + 1.89T + 73T^{2} \) |
| 79 | \( 1 - 4.11T + 79T^{2} \) |
| 83 | \( 1 - 2.69T + 83T^{2} \) |
| 89 | \( 1 + 1.81T + 89T^{2} \) |
| 97 | \( 1 + 14.1T + 97T^{2} \) |
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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
−7.86109971318887455728379562286, −7.05713682784764780898954232228, −6.68771348118003430199788020329, −5.52323949019628885401417084573, −4.31975266960774949635436348839, −3.81791014511952997111329186631, −2.72732035110406979302489026714, −2.06886016290099141354661346533, −1.28760100185724237196046579143, 0,
1.28760100185724237196046579143, 2.06886016290099141354661346533, 2.72732035110406979302489026714, 3.81791014511952997111329186631, 4.31975266960774949635436348839, 5.52323949019628885401417084573, 6.68771348118003430199788020329, 7.05713682784764780898954232228, 7.86109971318887455728379562286
