| L(s) = 1 | + 1.45·2-s + 0.120·4-s − 2.73·5-s − 4.41·7-s − 2.73·8-s − 3.98·10-s − 3.68·11-s − 0.758·13-s − 6.42·14-s − 4.22·16-s + 6.15·17-s − 0.330·20-s − 5.36·22-s − 0.505·23-s + 2.49·25-s − 1.10·26-s − 0.532·28-s − 3.18·29-s + 1.77·31-s − 0.681·32-s + 8.96·34-s + 12.0·35-s + 2.81·37-s + 7.49·40-s − 11.8·41-s + 2.30·43-s − 0.444·44-s + ⋯ |
| L(s) = 1 | + 1.02·2-s + 0.0603·4-s − 1.22·5-s − 1.66·7-s − 0.967·8-s − 1.26·10-s − 1.11·11-s − 0.210·13-s − 1.71·14-s − 1.05·16-s + 1.49·17-s − 0.0738·20-s − 1.14·22-s − 0.105·23-s + 0.498·25-s − 0.216·26-s − 0.100·28-s − 0.590·29-s + 0.318·31-s − 0.120·32-s + 1.53·34-s + 2.04·35-s + 0.462·37-s + 1.18·40-s − 1.84·41-s + 0.351·43-s − 0.0670·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8188949287\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8188949287\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 1.45T + 2T^{2} \) |
| 5 | \( 1 + 2.73T + 5T^{2} \) |
| 7 | \( 1 + 4.41T + 7T^{2} \) |
| 11 | \( 1 + 3.68T + 11T^{2} \) |
| 13 | \( 1 + 0.758T + 13T^{2} \) |
| 17 | \( 1 - 6.15T + 17T^{2} \) |
| 23 | \( 1 + 0.505T + 23T^{2} \) |
| 29 | \( 1 + 3.18T + 29T^{2} \) |
| 31 | \( 1 - 1.77T + 31T^{2} \) |
| 37 | \( 1 - 2.81T + 37T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 43 | \( 1 - 2.30T + 43T^{2} \) |
| 47 | \( 1 - 8.14T + 47T^{2} \) |
| 53 | \( 1 - 9.84T + 53T^{2} \) |
| 59 | \( 1 + 9.60T + 59T^{2} \) |
| 61 | \( 1 + 7.24T + 61T^{2} \) |
| 67 | \( 1 + 6.12T + 67T^{2} \) |
| 71 | \( 1 + 2.83T + 71T^{2} \) |
| 73 | \( 1 - 4.80T + 73T^{2} \) |
| 79 | \( 1 - 16.5T + 79T^{2} \) |
| 83 | \( 1 - 0.236T + 83T^{2} \) |
| 89 | \( 1 + 7.78T + 89T^{2} \) |
| 97 | \( 1 + 4.94T + 97T^{2} \) |
| 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
−8.579175163682764292946179431502, −7.73170331289889314711670976097, −7.15751346809225433726384188282, −6.16602889705350075964830802870, −5.57159358904416954270351064767, −4.73803216975477268541984654872, −3.77813490837862849657779462985, −3.35018493968626725169900029502, −2.66034493414734103708729874894, −0.43520512086109830230216209597,
0.43520512086109830230216209597, 2.66034493414734103708729874894, 3.35018493968626725169900029502, 3.77813490837862849657779462985, 4.73803216975477268541984654872, 5.57159358904416954270351064767, 6.16602889705350075964830802870, 7.15751346809225433726384188282, 7.73170331289889314711670976097, 8.579175163682764292946179431502
