| L(s) = 1 | + 3-s − 3.61·5-s + 3.30·7-s + 9-s + 3.59·11-s + 3.31·13-s − 3.61·15-s + 1.82·17-s + 7.34·19-s + 3.30·21-s + 1.18·23-s + 8.04·25-s + 27-s + 2.35·29-s + 3.67·31-s + 3.59·33-s − 11.9·35-s + 1.51·37-s + 3.31·39-s − 7.16·41-s − 2.14·43-s − 3.61·45-s − 4.46·47-s + 3.90·49-s + 1.82·51-s − 1.84·53-s − 12.9·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.61·5-s + 1.24·7-s + 0.333·9-s + 1.08·11-s + 0.920·13-s − 0.932·15-s + 0.442·17-s + 1.68·19-s + 0.720·21-s + 0.246·23-s + 1.60·25-s + 0.192·27-s + 0.436·29-s + 0.659·31-s + 0.625·33-s − 2.01·35-s + 0.249·37-s + 0.531·39-s − 1.11·41-s − 0.326·43-s − 0.538·45-s − 0.651·47-s + 0.557·49-s + 0.255·51-s − 0.252·53-s − 1.75·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.728581092\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.728581092\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 251 | \( 1 - T \) |
| good | 5 | \( 1 + 3.61T + 5T^{2} \) |
| 7 | \( 1 - 3.30T + 7T^{2} \) |
| 11 | \( 1 - 3.59T + 11T^{2} \) |
| 13 | \( 1 - 3.31T + 13T^{2} \) |
| 17 | \( 1 - 1.82T + 17T^{2} \) |
| 19 | \( 1 - 7.34T + 19T^{2} \) |
| 23 | \( 1 - 1.18T + 23T^{2} \) |
| 29 | \( 1 - 2.35T + 29T^{2} \) |
| 31 | \( 1 - 3.67T + 31T^{2} \) |
| 37 | \( 1 - 1.51T + 37T^{2} \) |
| 41 | \( 1 + 7.16T + 41T^{2} \) |
| 43 | \( 1 + 2.14T + 43T^{2} \) |
| 47 | \( 1 + 4.46T + 47T^{2} \) |
| 53 | \( 1 + 1.84T + 53T^{2} \) |
| 59 | \( 1 + 7.07T + 59T^{2} \) |
| 61 | \( 1 + 5.96T + 61T^{2} \) |
| 67 | \( 1 + 6.57T + 67T^{2} \) |
| 71 | \( 1 + 13.9T + 71T^{2} \) |
| 73 | \( 1 - 1.03T + 73T^{2} \) |
| 79 | \( 1 - 2.02T + 79T^{2} \) |
| 83 | \( 1 - 3.49T + 83T^{2} \) |
| 89 | \( 1 - 15.9T + 89T^{2} \) |
| 97 | \( 1 + 0.0910T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.023336382265424598542964610755, −7.59533986303671376112453851748, −6.93850058072553942995499527902, −5.99474204395060255816125300657, −4.83047530396681022069299352239, −4.47430529031659056333301078390, −3.46473360231362237338087995233, −3.21802690890909815902936460439, −1.61002453944713128723626229333, −0.950890135679982394378405237975,
0.950890135679982394378405237975, 1.61002453944713128723626229333, 3.21802690890909815902936460439, 3.46473360231362237338087995233, 4.47430529031659056333301078390, 4.83047530396681022069299352239, 5.99474204395060255816125300657, 6.93850058072553942995499527902, 7.59533986303671376112453851748, 8.023336382265424598542964610755
