| L(s) = 1 | − 0.112·2-s − 1.98·4-s − 1.06·5-s + 7-s + 0.448·8-s + 0.119·10-s + 5.84·13-s − 0.112·14-s + 3.92·16-s + 2.80·17-s + 3.56·19-s + 2.10·20-s − 4.72·23-s − 3.87·25-s − 0.657·26-s − 1.98·28-s + 8.59·29-s + 1.74·31-s − 1.33·32-s − 0.315·34-s − 1.06·35-s + 4.74·37-s − 0.400·38-s − 0.476·40-s + 6.13·41-s − 5.25·43-s + 0.531·46-s + ⋯ |
| L(s) = 1 | − 0.0795·2-s − 0.993·4-s − 0.474·5-s + 0.377·7-s + 0.158·8-s + 0.0377·10-s + 1.62·13-s − 0.0300·14-s + 0.981·16-s + 0.680·17-s + 0.817·19-s + 0.471·20-s − 0.984·23-s − 0.774·25-s − 0.128·26-s − 0.375·28-s + 1.59·29-s + 0.312·31-s − 0.236·32-s − 0.0541·34-s − 0.179·35-s + 0.780·37-s − 0.0649·38-s − 0.0752·40-s + 0.958·41-s − 0.800·43-s + 0.0783·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.637607799\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.637607799\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.112T + 2T^{2} \) |
| 5 | \( 1 + 1.06T + 5T^{2} \) |
| 13 | \( 1 - 5.84T + 13T^{2} \) |
| 17 | \( 1 - 2.80T + 17T^{2} \) |
| 19 | \( 1 - 3.56T + 19T^{2} \) |
| 23 | \( 1 + 4.72T + 23T^{2} \) |
| 29 | \( 1 - 8.59T + 29T^{2} \) |
| 31 | \( 1 - 1.74T + 31T^{2} \) |
| 37 | \( 1 - 4.74T + 37T^{2} \) |
| 41 | \( 1 - 6.13T + 41T^{2} \) |
| 43 | \( 1 + 5.25T + 43T^{2} \) |
| 47 | \( 1 - 8.19T + 47T^{2} \) |
| 53 | \( 1 + 14.3T + 53T^{2} \) |
| 59 | \( 1 + 3.90T + 59T^{2} \) |
| 61 | \( 1 - 14.7T + 61T^{2} \) |
| 67 | \( 1 - 8.29T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 - 4.20T + 73T^{2} \) |
| 79 | \( 1 + 6.80T + 79T^{2} \) |
| 83 | \( 1 - 1.16T + 83T^{2} \) |
| 89 | \( 1 + 2.90T + 89T^{2} \) |
| 97 | \( 1 + 4.85T + 97T^{2} \) |
| 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
−8.146632563223717900105317271088, −7.43670030396318820704688488452, −6.31419479338446808692093859060, −5.79202938525771988675501666295, −5.00579444300185450123368305643, −4.18345990965898966723455861217, −3.74882840908614131891644268952, −2.88688608481253893440035429356, −1.46795974368203034391901206849, −0.72702679855630541874826691832,
0.72702679855630541874826691832, 1.46795974368203034391901206849, 2.88688608481253893440035429356, 3.74882840908614131891644268952, 4.18345990965898966723455861217, 5.00579444300185450123368305643, 5.79202938525771988675501666295, 6.31419479338446808692093859060, 7.43670030396318820704688488452, 8.146632563223717900105317271088
