| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s − 4.13·5-s − 6·6-s + 4.92·7-s + 8·8-s + 9·9-s − 8.27·10-s − 59.9·11-s − 12·12-s + 45.6·13-s + 9.84·14-s + 12.4·15-s + 16·16-s − 95.3·17-s + 18·18-s + 27.3·19-s − 16.5·20-s − 14.7·21-s − 119.·22-s − 43.2·23-s − 24·24-s − 107.·25-s + 91.3·26-s − 27·27-s + 19.6·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.369·5-s − 0.408·6-s + 0.265·7-s + 0.353·8-s + 0.333·9-s − 0.261·10-s − 1.64·11-s − 0.288·12-s + 0.974·13-s + 0.188·14-s + 0.213·15-s + 0.250·16-s − 1.36·17-s + 0.235·18-s + 0.330·19-s − 0.184·20-s − 0.153·21-s − 1.16·22-s − 0.392·23-s − 0.204·24-s − 0.863·25-s + 0.688·26-s − 0.192·27-s + 0.132·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 + 3T \) |
| 59 | \( 1 + 59T \) |
| good | 5 | \( 1 + 4.13T + 125T^{2} \) |
| 7 | \( 1 - 4.92T + 343T^{2} \) |
| 11 | \( 1 + 59.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 45.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 95.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 27.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 43.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 133.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 67.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 309.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 124.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 22.2T + 7.95e4T^{2} \) |
| 47 | \( 1 + 267.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 94.7T + 1.48e5T^{2} \) |
| 61 | \( 1 + 39.2T + 2.26e5T^{2} \) |
| 67 | \( 1 - 127.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 536.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 472.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 504.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 77.1T + 5.71e5T^{2} \) |
| 89 | \( 1 - 460.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 678.T + 9.12e5T^{2} \) |
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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
−10.95229215272000870490163086206, −9.971760911896839538622990145787, −8.481391767102000889326562233363, −7.63037193802697110669845239741, −6.53189874202142220596010457773, −5.52018106740490800963172836019, −4.67208816331478322928612442765, −3.49004442768789641355430950336, −1.98297379574405076970237297158, 0,
1.98297379574405076970237297158, 3.49004442768789641355430950336, 4.67208816331478322928612442765, 5.52018106740490800963172836019, 6.53189874202142220596010457773, 7.63037193802697110669845239741, 8.481391767102000889326562233363, 9.971760911896839538622990145787, 10.95229215272000870490163086206
