| L(s) = 1 | + 2·2-s + 4·4-s − 12.7·5-s − 1.26·7-s + 8·8-s − 25.4·10-s − 13.2·13-s − 2.53·14-s + 16·16-s − 20.9·17-s − 88.9·19-s − 50.9·20-s + 38.1·23-s + 37.1·25-s − 26.5·26-s − 5.06·28-s − 16.5·29-s + 165.·31-s + 32·32-s − 41.8·34-s + 16.1·35-s + 252.·37-s − 177.·38-s − 101.·40-s − 328.·41-s − 429.·43-s + 76.3·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.13·5-s − 0.0684·7-s + 0.353·8-s − 0.805·10-s − 0.283·13-s − 0.0483·14-s + 0.250·16-s − 0.298·17-s − 1.07·19-s − 0.569·20-s + 0.346·23-s + 0.297·25-s − 0.200·26-s − 0.0342·28-s − 0.105·29-s + 0.958·31-s + 0.176·32-s − 0.211·34-s + 0.0779·35-s + 1.12·37-s − 0.759·38-s − 0.402·40-s − 1.25·41-s − 1.52·43-s + 0.244·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.116331554\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.116331554\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 12.7T + 125T^{2} \) |
| 7 | \( 1 + 1.26T + 343T^{2} \) |
| 13 | \( 1 + 13.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 20.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 88.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 38.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 16.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 165.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 252.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 328.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 429.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 240.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 461.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 313.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 476.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 510.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 232.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 511.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 681.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 919.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 120.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 426.T + 9.12e5T^{2} \) |
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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
−8.389063399862477845687919451551, −8.013491920954027775059814651623, −6.97036024973709379379807853387, −6.50625498025833839913456287822, −5.38369894986244294267655683096, −4.54236579933278955871034168501, −3.92972152074078867014411848940, −3.05424356902951356470320340061, −2.03091900883497612843767648399, −0.57128264475422727278105602075,
0.57128264475422727278105602075, 2.03091900883497612843767648399, 3.05424356902951356470320340061, 3.92972152074078867014411848940, 4.54236579933278955871034168501, 5.38369894986244294267655683096, 6.50625498025833839913456287822, 6.97036024973709379379807853387, 8.013491920954027775059814651623, 8.389063399862477845687919451551
