| L(s) = 1 | − 2.69·2-s − 2.04·3-s + 5.25·4-s − 1.47·5-s + 5.51·6-s − 4.98·7-s − 8.75·8-s + 1.18·9-s + 3.96·10-s + 1.63·11-s − 10.7·12-s + 0.394·13-s + 13.4·14-s + 3.01·15-s + 13.0·16-s − 1.95·17-s − 3.19·18-s + 2.39·19-s − 7.72·20-s + 10.2·21-s − 4.40·22-s + 6.20·23-s + 17.9·24-s − 2.83·25-s − 1.06·26-s + 3.70·27-s − 26.1·28-s + ⋯ |
| L(s) = 1 | − 1.90·2-s − 1.18·3-s + 2.62·4-s − 0.657·5-s + 2.24·6-s − 1.88·7-s − 3.09·8-s + 0.396·9-s + 1.25·10-s + 0.492·11-s − 3.10·12-s + 0.109·13-s + 3.58·14-s + 0.777·15-s + 3.26·16-s − 0.474·17-s − 0.753·18-s + 0.550·19-s − 1.72·20-s + 2.22·21-s − 0.938·22-s + 1.29·23-s + 3.65·24-s − 0.567·25-s − 0.208·26-s + 0.713·27-s − 4.94·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9409 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9409 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2408899676\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2408899676\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 97 | \( 1 \) |
| good | 2 | \( 1 + 2.69T + 2T^{2} \) |
| 3 | \( 1 + 2.04T + 3T^{2} \) |
| 5 | \( 1 + 1.47T + 5T^{2} \) |
| 7 | \( 1 + 4.98T + 7T^{2} \) |
| 11 | \( 1 - 1.63T + 11T^{2} \) |
| 13 | \( 1 - 0.394T + 13T^{2} \) |
| 17 | \( 1 + 1.95T + 17T^{2} \) |
| 19 | \( 1 - 2.39T + 19T^{2} \) |
| 23 | \( 1 - 6.20T + 23T^{2} \) |
| 29 | \( 1 - 0.717T + 29T^{2} \) |
| 31 | \( 1 + 3.12T + 31T^{2} \) |
| 37 | \( 1 - 5.68T + 37T^{2} \) |
| 41 | \( 1 - 5.97T + 41T^{2} \) |
| 43 | \( 1 + 0.777T + 43T^{2} \) |
| 47 | \( 1 - 4.88T + 47T^{2} \) |
| 53 | \( 1 + 6.68T + 53T^{2} \) |
| 59 | \( 1 + 0.201T + 59T^{2} \) |
| 61 | \( 1 - 9.97T + 61T^{2} \) |
| 67 | \( 1 - 0.930T + 67T^{2} \) |
| 71 | \( 1 - 13.1T + 71T^{2} \) |
| 73 | \( 1 - 11.4T + 73T^{2} \) |
| 79 | \( 1 - 9.92T + 79T^{2} \) |
| 83 | \( 1 + 15.0T + 83T^{2} \) |
| 89 | \( 1 + 6.38T + 89T^{2} \) |
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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
−7.63315175083448793552621961811, −6.99792402957172601713549820304, −6.57886371072209003354468022209, −6.08506835868848480408647396799, −5.34853401786305912624181029219, −3.99200529300302517463697969249, −3.18708165464177483477352820948, −2.44216058820140041813585619798, −1.00958880072679793055003353670, −0.42753728511484825966891711667,
0.42753728511484825966891711667, 1.00958880072679793055003353670, 2.44216058820140041813585619798, 3.18708165464177483477352820948, 3.99200529300302517463697969249, 5.34853401786305912624181029219, 6.08506835868848480408647396799, 6.57886371072209003354468022209, 6.99792402957172601713549820304, 7.63315175083448793552621961811
