| L(s) = 1 | − 0.520·2-s − 3.35·3-s − 1.72·4-s + 5-s + 1.74·6-s + 7-s + 1.94·8-s + 8.24·9-s − 0.520·10-s + 0.0909·11-s + 5.79·12-s + 3.30·13-s − 0.520·14-s − 3.35·15-s + 2.44·16-s + 1.15·17-s − 4.29·18-s + 4.14·19-s − 1.72·20-s − 3.35·21-s − 0.0473·22-s + 6.21·23-s − 6.51·24-s + 25-s − 1.72·26-s − 17.5·27-s − 1.72·28-s + ⋯ |
| L(s) = 1 | − 0.368·2-s − 1.93·3-s − 0.864·4-s + 0.447·5-s + 0.712·6-s + 0.377·7-s + 0.686·8-s + 2.74·9-s − 0.164·10-s + 0.0274·11-s + 1.67·12-s + 0.917·13-s − 0.139·14-s − 0.865·15-s + 0.611·16-s + 0.279·17-s − 1.01·18-s + 0.950·19-s − 0.386·20-s − 0.731·21-s − 0.0100·22-s + 1.29·23-s − 1.32·24-s + 0.200·25-s − 0.337·26-s − 3.38·27-s − 0.326·28-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 | \( 1 + 0.520T + 2T^{2} \) |
| 3 | \( 1 + 3.35T + 3T^{2} \) |
| 11 | \( 1 - 0.0909T + 11T^{2} \) |
| 13 | \( 1 - 3.30T + 13T^{2} \) |
| 17 | \( 1 - 1.15T + 17T^{2} \) |
| 19 | \( 1 - 4.14T + 19T^{2} \) |
| 23 | \( 1 - 6.21T + 23T^{2} \) |
| 29 | \( 1 + 2.56T + 29T^{2} \) |
| 31 | \( 1 - 1.72T + 31T^{2} \) |
| 37 | \( 1 + 4.36T + 37T^{2} \) |
| 41 | \( 1 + 7.98T + 41T^{2} \) |
| 43 | \( 1 + 8.73T + 43T^{2} \) |
| 47 | \( 1 - 5.90T + 47T^{2} \) |
| 53 | \( 1 + 3.00T + 53T^{2} \) |
| 59 | \( 1 + 5.95T + 59T^{2} \) |
| 61 | \( 1 + 4.63T + 61T^{2} \) |
| 67 | \( 1 - 4.50T + 67T^{2} \) |
| 71 | \( 1 + 3.82T + 71T^{2} \) |
| 73 | \( 1 + 3.56T + 73T^{2} \) |
| 79 | \( 1 + 4.66T + 79T^{2} \) |
| 83 | \( 1 - 5.74T + 83T^{2} \) |
| 89 | \( 1 + 17.2T + 89T^{2} \) |
| 97 | \( 1 + 3.41T + 97T^{2} \) |
| 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
−7.31512183289479029970712720351, −6.75071980531296427865351300233, −5.95117036967182580799990609599, −5.27873103364081505847979557009, −5.02350337084141490913555319032, −4.19299958737072461318149544863, −3.33451004809170094652193576329, −1.50487826282061139759019652047, −1.14945688365789230879877270897, 0,
1.14945688365789230879877270897, 1.50487826282061139759019652047, 3.33451004809170094652193576329, 4.19299958737072461318149544863, 5.02350337084141490913555319032, 5.27873103364081505847979557009, 5.95117036967182580799990609599, 6.75071980531296427865351300233, 7.31512183289479029970712720351
