| L(s) = 1 | − 2-s + 0.325·3-s + 4-s − 2.89·5-s − 0.325·6-s + 0.0543·7-s − 8-s − 2.89·9-s + 2.89·10-s − 0.662·11-s + 0.325·12-s + 1.69·13-s − 0.0543·14-s − 0.943·15-s + 16-s − 5.74·17-s + 2.89·18-s − 6.93·19-s − 2.89·20-s + 0.0176·21-s + 0.662·22-s − 23-s − 0.325·24-s + 3.40·25-s − 1.69·26-s − 1.91·27-s + 0.0543·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.187·3-s + 0.5·4-s − 1.29·5-s − 0.132·6-s + 0.0205·7-s − 0.353·8-s − 0.964·9-s + 0.917·10-s − 0.199·11-s + 0.0939·12-s + 0.471·13-s − 0.0145·14-s − 0.243·15-s + 0.250·16-s − 1.39·17-s + 0.682·18-s − 1.59·19-s − 0.648·20-s + 0.00385·21-s + 0.141·22-s − 0.208·23-s − 0.0664·24-s + 0.681·25-s − 0.333·26-s − 0.369·27-s + 0.0102·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2532367371\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2532367371\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 - 0.325T + 3T^{2} \) |
| 5 | \( 1 + 2.89T + 5T^{2} \) |
| 7 | \( 1 - 0.0543T + 7T^{2} \) |
| 11 | \( 1 + 0.662T + 11T^{2} \) |
| 13 | \( 1 - 1.69T + 13T^{2} \) |
| 17 | \( 1 + 5.74T + 17T^{2} \) |
| 19 | \( 1 + 6.93T + 19T^{2} \) |
| 29 | \( 1 + 8.08T + 29T^{2} \) |
| 31 | \( 1 - 2.69T + 31T^{2} \) |
| 37 | \( 1 - 3.33T + 37T^{2} \) |
| 41 | \( 1 + 3.97T + 41T^{2} \) |
| 43 | \( 1 + 5.40T + 43T^{2} \) |
| 47 | \( 1 + 0.462T + 47T^{2} \) |
| 53 | \( 1 - 7.11T + 53T^{2} \) |
| 59 | \( 1 + 5.40T + 59T^{2} \) |
| 61 | \( 1 - 3.99T + 61T^{2} \) |
| 67 | \( 1 + 4.52T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 - 1.69T + 73T^{2} \) |
| 79 | \( 1 - 3.11T + 79T^{2} \) |
| 83 | \( 1 + 2.35T + 83T^{2} \) |
| 89 | \( 1 + 6.29T + 89T^{2} \) |
| 97 | \( 1 + 4.56T + 97T^{2} \) |
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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
−8.175402268014235667397393859980, −7.64406374385649317950078764521, −6.70534333447054805052808820986, −6.24503073660152522979107814777, −5.20633636656565881759014082841, −4.22383743827770335886538279375, −3.66138106713853558990245072576, −2.68544819003275771211401517973, −1.86687924623413326194767442536, −0.27396572988660052135742897356,
0.27396572988660052135742897356, 1.86687924623413326194767442536, 2.68544819003275771211401517973, 3.66138106713853558990245072576, 4.22383743827770335886538279375, 5.20633636656565881759014082841, 6.24503073660152522979107814777, 6.70534333447054805052808820986, 7.64406374385649317950078764521, 8.175402268014235667397393859980
