| L(s) = 1 | + 0.582·2-s − 0.306·3-s − 1.66·4-s − 5-s − 0.178·6-s − 3.02·7-s − 2.13·8-s − 2.90·9-s − 0.582·10-s + 0.508·12-s − 13-s − 1.76·14-s + 0.306·15-s + 2.07·16-s + 0.657·17-s − 1.69·18-s + 1.08·19-s + 1.66·20-s + 0.925·21-s + 0.839·23-s + 0.653·24-s + 25-s − 0.582·26-s + 1.80·27-s + 5.02·28-s + 9.63·29-s + 0.178·30-s + ⋯ |
| L(s) = 1 | + 0.412·2-s − 0.176·3-s − 0.830·4-s − 0.447·5-s − 0.0728·6-s − 1.14·7-s − 0.754·8-s − 0.968·9-s − 0.184·10-s + 0.146·12-s − 0.277·13-s − 0.470·14-s + 0.0790·15-s + 0.519·16-s + 0.159·17-s − 0.399·18-s + 0.249·19-s + 0.371·20-s + 0.202·21-s + 0.175·23-s + 0.133·24-s + 0.200·25-s − 0.114·26-s + 0.348·27-s + 0.948·28-s + 1.78·29-s + 0.0325·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7865 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7865 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 0.582T + 2T^{2} \) |
| 3 | \( 1 + 0.306T + 3T^{2} \) |
| 7 | \( 1 + 3.02T + 7T^{2} \) |
| 17 | \( 1 - 0.657T + 17T^{2} \) |
| 19 | \( 1 - 1.08T + 19T^{2} \) |
| 23 | \( 1 - 0.839T + 23T^{2} \) |
| 29 | \( 1 - 9.63T + 29T^{2} \) |
| 31 | \( 1 - 4.46T + 31T^{2} \) |
| 37 | \( 1 + 0.992T + 37T^{2} \) |
| 41 | \( 1 + 7.35T + 41T^{2} \) |
| 43 | \( 1 - 5.45T + 43T^{2} \) |
| 47 | \( 1 - 6.29T + 47T^{2} \) |
| 53 | \( 1 - 14.3T + 53T^{2} \) |
| 59 | \( 1 + 12.0T + 59T^{2} \) |
| 61 | \( 1 - 3.03T + 61T^{2} \) |
| 67 | \( 1 + 3.16T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 + 9.88T + 73T^{2} \) |
| 79 | \( 1 - 11.8T + 79T^{2} \) |
| 83 | \( 1 + 8.89T + 83T^{2} \) |
| 89 | \( 1 + 0.711T + 89T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.47594840438233058389567330290, −6.63313906367731851169410670297, −6.04706061180386612688370756902, −5.39049203971847298645208188086, −4.65865745921626221651148958958, −3.95007493209343027537850435177, −3.07406508437826796535512595894, −2.74529892065399809570490988706, −0.900249863778393490350753966585, 0,
0.900249863778393490350753966585, 2.74529892065399809570490988706, 3.07406508437826796535512595894, 3.95007493209343027537850435177, 4.65865745921626221651148958958, 5.39049203971847298645208188086, 6.04706061180386612688370756902, 6.63313906367731851169410670297, 7.47594840438233058389567330290
