| L(s) = 1 | + 1.84·3-s − 5-s − 0.0636·7-s + 0.394·9-s + 0.656·11-s − 6.07·13-s − 1.84·15-s + 7.76·17-s + 1.00·19-s − 0.117·21-s + 8.16·23-s + 25-s − 4.79·27-s − 8.67·29-s + 2.13·31-s + 1.20·33-s + 0.0636·35-s + 6.25·37-s − 11.1·39-s + 0.177·41-s − 4.30·43-s − 0.394·45-s + 11.4·47-s − 6.99·49-s + 14.3·51-s − 4.81·53-s − 0.656·55-s + ⋯ |
| L(s) = 1 | + 1.06·3-s − 0.447·5-s − 0.0240·7-s + 0.131·9-s + 0.197·11-s − 1.68·13-s − 0.475·15-s + 1.88·17-s + 0.230·19-s − 0.0256·21-s + 1.70·23-s + 0.200·25-s − 0.923·27-s − 1.61·29-s + 0.384·31-s + 0.210·33-s + 0.0107·35-s + 1.02·37-s − 1.79·39-s + 0.0276·41-s − 0.657·43-s − 0.0588·45-s + 1.66·47-s − 0.999·49-s + 2.00·51-s − 0.661·53-s − 0.0884·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.564286575\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.564286575\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 3 | \( 1 - 1.84T + 3T^{2} \) |
| 7 | \( 1 + 0.0636T + 7T^{2} \) |
| 11 | \( 1 - 0.656T + 11T^{2} \) |
| 13 | \( 1 + 6.07T + 13T^{2} \) |
| 17 | \( 1 - 7.76T + 17T^{2} \) |
| 19 | \( 1 - 1.00T + 19T^{2} \) |
| 23 | \( 1 - 8.16T + 23T^{2} \) |
| 29 | \( 1 + 8.67T + 29T^{2} \) |
| 31 | \( 1 - 2.13T + 31T^{2} \) |
| 37 | \( 1 - 6.25T + 37T^{2} \) |
| 41 | \( 1 - 0.177T + 41T^{2} \) |
| 43 | \( 1 + 4.30T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 + 4.81T + 53T^{2} \) |
| 59 | \( 1 + 9.27T + 59T^{2} \) |
| 61 | \( 1 - 10.7T + 61T^{2} \) |
| 67 | \( 1 - 14.8T + 67T^{2} \) |
| 71 | \( 1 + 2.12T + 71T^{2} \) |
| 73 | \( 1 - 0.427T + 73T^{2} \) |
| 79 | \( 1 - 10.0T + 79T^{2} \) |
| 83 | \( 1 - 3.90T + 83T^{2} \) |
| 89 | \( 1 + 7.59T + 89T^{2} \) |
| 97 | \( 1 - 0.147T + 97T^{2} \) |
| 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
−7.64013782739872940955700653441, −7.55446034234827040373726595907, −6.68429114972797944965694970001, −5.53511182376162131443676008118, −5.09428335881489703172395541858, −4.12201316450915779959069742800, −3.28315947964582729824549615145, −2.88338571547048309660782266678, −1.95430613219558164864880014307, −0.74317528758215143332881261758,
0.74317528758215143332881261758, 1.95430613219558164864880014307, 2.88338571547048309660782266678, 3.28315947964582729824549615145, 4.12201316450915779959069742800, 5.09428335881489703172395541858, 5.53511182376162131443676008118, 6.68429114972797944965694970001, 7.55446034234827040373726595907, 7.64013782739872940955700653441
