| L(s) = 1 | + 2-s + 4-s + 1.80·5-s − 3.56·7-s + 8-s + 1.80·10-s − 4.98·11-s + 5.16·13-s − 3.56·14-s + 16-s − 6.45·17-s + 7.14·19-s + 1.80·20-s − 4.98·22-s + 8.13·23-s − 1.75·25-s + 5.16·26-s − 3.56·28-s − 2.46·29-s + 4.60·31-s + 32-s − 6.45·34-s − 6.42·35-s + 3.27·37-s + 7.14·38-s + 1.80·40-s − 2.73·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.805·5-s − 1.34·7-s + 0.353·8-s + 0.569·10-s − 1.50·11-s + 1.43·13-s − 0.953·14-s + 0.250·16-s − 1.56·17-s + 1.63·19-s + 0.402·20-s − 1.06·22-s + 1.69·23-s − 0.351·25-s + 1.01·26-s − 0.674·28-s − 0.456·29-s + 0.827·31-s + 0.176·32-s − 1.10·34-s − 1.08·35-s + 0.537·37-s + 1.15·38-s + 0.284·40-s − 0.426·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.988202026\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.988202026\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 223 | \( 1 - T \) |
| good | 5 | \( 1 - 1.80T + 5T^{2} \) |
| 7 | \( 1 + 3.56T + 7T^{2} \) |
| 11 | \( 1 + 4.98T + 11T^{2} \) |
| 13 | \( 1 - 5.16T + 13T^{2} \) |
| 17 | \( 1 + 6.45T + 17T^{2} \) |
| 19 | \( 1 - 7.14T + 19T^{2} \) |
| 23 | \( 1 - 8.13T + 23T^{2} \) |
| 29 | \( 1 + 2.46T + 29T^{2} \) |
| 31 | \( 1 - 4.60T + 31T^{2} \) |
| 37 | \( 1 - 3.27T + 37T^{2} \) |
| 41 | \( 1 + 2.73T + 41T^{2} \) |
| 43 | \( 1 - 11.7T + 43T^{2} \) |
| 47 | \( 1 - 10.8T + 47T^{2} \) |
| 53 | \( 1 - 1.54T + 53T^{2} \) |
| 59 | \( 1 - 4.79T + 59T^{2} \) |
| 61 | \( 1 - 3.38T + 61T^{2} \) |
| 67 | \( 1 + 2.06T + 67T^{2} \) |
| 71 | \( 1 + 6.68T + 71T^{2} \) |
| 73 | \( 1 - 10.6T + 73T^{2} \) |
| 79 | \( 1 + 5.51T + 79T^{2} \) |
| 83 | \( 1 - 3.69T + 83T^{2} \) |
| 89 | \( 1 - 3.29T + 89T^{2} \) |
| 97 | \( 1 - 3.80T + 97T^{2} \) |
| 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
−8.528902758393782222286967836717, −7.43750293537567631166421753729, −6.86392781498594808649221236695, −5.99214777480665371382011560423, −5.66808759306294629265165786265, −4.77532822934103630039519212899, −3.73702446155632634237112569418, −2.92744210007617722298771774712, −2.36096224770803454338912785737, −0.886907967043702912198960064189,
0.886907967043702912198960064189, 2.36096224770803454338912785737, 2.92744210007617722298771774712, 3.73702446155632634237112569418, 4.77532822934103630039519212899, 5.66808759306294629265165786265, 5.99214777480665371382011560423, 6.86392781498594808649221236695, 7.43750293537567631166421753729, 8.528902758393782222286967836717
