| L(s) = 1 | + 3-s − 2.76·7-s + 9-s + 4.49·11-s − 13-s − 3.62·17-s − 6.14·19-s − 2.76·21-s − 5.52·23-s + 27-s + 6.25·29-s + 8.49·31-s + 4.49·33-s + 0.896·37-s − 39-s + 2.89·41-s − 0.270·43-s − 3.38·47-s + 0.626·49-s − 3.62·51-s + 3.27·53-s − 6.14·57-s − 9.53·59-s + 11.4·61-s − 2.76·63-s + 12.4·67-s − 5.52·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.04·7-s + 0.333·9-s + 1.35·11-s − 0.277·13-s − 0.879·17-s − 1.41·19-s − 0.602·21-s − 1.15·23-s + 0.192·27-s + 1.16·29-s + 1.52·31-s + 0.781·33-s + 0.147·37-s − 0.160·39-s + 0.452·41-s − 0.0412·43-s − 0.494·47-s + 0.0894·49-s − 0.507·51-s + 0.449·53-s − 0.814·57-s − 1.24·59-s + 1.46·61-s − 0.347·63-s + 1.51·67-s − 0.664·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.057588496\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.057588496\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 2.76T + 7T^{2} \) |
| 11 | \( 1 - 4.49T + 11T^{2} \) |
| 17 | \( 1 + 3.62T + 17T^{2} \) |
| 19 | \( 1 + 6.14T + 19T^{2} \) |
| 23 | \( 1 + 5.52T + 23T^{2} \) |
| 29 | \( 1 - 6.25T + 29T^{2} \) |
| 31 | \( 1 - 8.49T + 31T^{2} \) |
| 37 | \( 1 - 0.896T + 37T^{2} \) |
| 41 | \( 1 - 2.89T + 41T^{2} \) |
| 43 | \( 1 + 0.270T + 43T^{2} \) |
| 47 | \( 1 + 3.38T + 47T^{2} \) |
| 53 | \( 1 - 3.27T + 53T^{2} \) |
| 59 | \( 1 + 9.53T + 59T^{2} \) |
| 61 | \( 1 - 11.4T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 - 8.89T + 71T^{2} \) |
| 73 | \( 1 + 3.25T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 - 3.03T + 83T^{2} \) |
| 89 | \( 1 + 2.20T + 89T^{2} \) |
| 97 | \( 1 - 17.7T + 97T^{2} \) |
| 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
−8.066827691633452624584585035174, −6.95226843690636668021618399339, −6.41796797197110059155321250359, −6.21762979626372457683138029856, −4.81300769640275445846210166649, −4.16442458450561634636238104478, −3.60212356180885519765312607053, −2.63587982411024090549093492802, −1.97163636720051956428110036566, −0.67744902604118656560256861552,
0.67744902604118656560256861552, 1.97163636720051956428110036566, 2.63587982411024090549093492802, 3.60212356180885519765312607053, 4.16442458450561634636238104478, 4.81300769640275445846210166649, 6.21762979626372457683138029856, 6.41796797197110059155321250359, 6.95226843690636668021618399339, 8.066827691633452624584585035174
