| L(s) = 1 | − 2-s + 3-s + 4-s + 0.826·5-s − 6-s + 3.49·7-s − 8-s + 9-s − 0.826·10-s + 12-s + 13-s − 3.49·14-s + 0.826·15-s + 16-s − 7.15·17-s − 18-s + 2.09·19-s + 0.826·20-s + 3.49·21-s − 7.22·23-s − 24-s − 4.31·25-s − 26-s + 27-s + 3.49·28-s + 5.79·29-s − 0.826·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.369·5-s − 0.408·6-s + 1.32·7-s − 0.353·8-s + 0.333·9-s − 0.261·10-s + 0.288·12-s + 0.277·13-s − 0.934·14-s + 0.213·15-s + 0.250·16-s − 1.73·17-s − 0.235·18-s + 0.480·19-s + 0.184·20-s + 0.762·21-s − 1.50·23-s − 0.204·24-s − 0.863·25-s − 0.196·26-s + 0.192·27-s + 0.660·28-s + 1.07·29-s − 0.150·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9438 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9438 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 0.826T + 5T^{2} \) |
| 7 | \( 1 - 3.49T + 7T^{2} \) |
| 17 | \( 1 + 7.15T + 17T^{2} \) |
| 19 | \( 1 - 2.09T + 19T^{2} \) |
| 23 | \( 1 + 7.22T + 23T^{2} \) |
| 29 | \( 1 - 5.79T + 29T^{2} \) |
| 31 | \( 1 + 5.74T + 31T^{2} \) |
| 37 | \( 1 + 1.19T + 37T^{2} \) |
| 41 | \( 1 + 7.14T + 41T^{2} \) |
| 43 | \( 1 + 1.71T + 43T^{2} \) |
| 47 | \( 1 + 9.35T + 47T^{2} \) |
| 53 | \( 1 + 0.952T + 53T^{2} \) |
| 59 | \( 1 - 0.548T + 59T^{2} \) |
| 61 | \( 1 + 7.03T + 61T^{2} \) |
| 67 | \( 1 + 7.68T + 67T^{2} \) |
| 71 | \( 1 + 11.5T + 71T^{2} \) |
| 73 | \( 1 - 1.52T + 73T^{2} \) |
| 79 | \( 1 - 17.6T + 79T^{2} \) |
| 83 | \( 1 + 1.61T + 83T^{2} \) |
| 89 | \( 1 - 4.60T + 89T^{2} \) |
| 97 | \( 1 + 14.5T + 97T^{2} \) |
| 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
−7.59021761900488247259888487649, −6.79021718404184291598386104076, −6.17828100297226060506285692641, −5.28186218716251302344535511103, −4.54402736969252631428840319240, −3.82584197090739636489595447916, −2.77048733893305606156246548543, −1.85046896998338975224259412122, −1.60259938896505795183473695802, 0,
1.60259938896505795183473695802, 1.85046896998338975224259412122, 2.77048733893305606156246548543, 3.82584197090739636489595447916, 4.54402736969252631428840319240, 5.28186218716251302344535511103, 6.17828100297226060506285692641, 6.79021718404184291598386104076, 7.59021761900488247259888487649
