| L(s) = 1 | − 5-s − 3.05·7-s + 0.259·11-s + 2.79·13-s − 17-s + 6.10·19-s − 7.57·23-s + 25-s + 2.51·29-s + 10.3·31-s + 3.05·35-s − 8.10·37-s + 0.102·41-s − 1.20·43-s − 3.31·47-s + 2.31·49-s − 12.6·53-s − 0.259·55-s + 3.48·59-s − 14.2·61-s − 2.79·65-s − 4.68·67-s − 3.74·71-s + 10.7·73-s − 0.791·77-s − 3.74·79-s + 11.8·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.15·7-s + 0.0782·11-s + 0.774·13-s − 0.242·17-s + 1.40·19-s − 1.57·23-s + 0.200·25-s + 0.467·29-s + 1.86·31-s + 0.515·35-s − 1.33·37-s + 0.0160·41-s − 0.184·43-s − 0.482·47-s + 0.330·49-s − 1.73·53-s − 0.0349·55-s + 0.453·59-s − 1.81·61-s − 0.346·65-s − 0.572·67-s − 0.443·71-s + 1.25·73-s − 0.0902·77-s − 0.420·79-s + 1.29·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + 3.05T + 7T^{2} \) |
| 11 | \( 1 - 0.259T + 11T^{2} \) |
| 13 | \( 1 - 2.79T + 13T^{2} \) |
| 19 | \( 1 - 6.10T + 19T^{2} \) |
| 23 | \( 1 + 7.57T + 23T^{2} \) |
| 29 | \( 1 - 2.51T + 29T^{2} \) |
| 31 | \( 1 - 10.3T + 31T^{2} \) |
| 37 | \( 1 + 8.10T + 37T^{2} \) |
| 41 | \( 1 - 0.102T + 41T^{2} \) |
| 43 | \( 1 + 1.20T + 43T^{2} \) |
| 47 | \( 1 + 3.31T + 47T^{2} \) |
| 53 | \( 1 + 12.6T + 53T^{2} \) |
| 59 | \( 1 - 3.48T + 59T^{2} \) |
| 61 | \( 1 + 14.2T + 61T^{2} \) |
| 67 | \( 1 + 4.68T + 67T^{2} \) |
| 71 | \( 1 + 3.74T + 71T^{2} \) |
| 73 | \( 1 - 10.7T + 73T^{2} \) |
| 79 | \( 1 + 3.74T + 79T^{2} \) |
| 83 | \( 1 - 11.8T + 83T^{2} \) |
| 89 | \( 1 + 16.8T + 89T^{2} \) |
| 97 | \( 1 - 3.03T + 97T^{2} \) |
| 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
−8.246748384358894170722282336624, −7.70563275289504926223995268910, −6.60567558247877298519916852416, −6.30884778587964943900503051884, −5.29308555127652071847804786432, −4.29650825232462895413752411555, −3.46871737111990655977667617866, −2.84132639512746817217724651946, −1.37809612212633157396513299839, 0,
1.37809612212633157396513299839, 2.84132639512746817217724651946, 3.46871737111990655977667617866, 4.29650825232462895413752411555, 5.29308555127652071847804786432, 6.30884778587964943900503051884, 6.60567558247877298519916852416, 7.70563275289504926223995268910, 8.246748384358894170722282336624
