| L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s − 15.4·5-s + 6·6-s − 16.9·7-s − 8·8-s + 9·9-s + 30.8·10-s + 11·11-s − 12·12-s − 50.7·13-s + 33.8·14-s + 46.2·15-s + 16·16-s + 38.1·17-s − 18·18-s − 76.6·19-s − 61.6·20-s + 50.8·21-s − 22·22-s − 134.·23-s + 24·24-s + 112.·25-s + 101.·26-s − 27·27-s − 67.7·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.37·5-s + 0.408·6-s − 0.914·7-s − 0.353·8-s + 0.333·9-s + 0.974·10-s + 0.301·11-s − 0.288·12-s − 1.08·13-s + 0.646·14-s + 0.795·15-s + 0.250·16-s + 0.543·17-s − 0.235·18-s − 0.925·19-s − 0.689·20-s + 0.528·21-s − 0.213·22-s − 1.21·23-s + 0.204·24-s + 0.899·25-s + 0.765·26-s − 0.192·27-s − 0.457·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2442 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2442 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 3T \) |
| 11 | \( 1 - 11T \) |
| 37 | \( 1 + 37T \) |
| good | 5 | \( 1 + 15.4T + 125T^{2} \) |
| 7 | \( 1 + 16.9T + 343T^{2} \) |
| 13 | \( 1 + 50.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 38.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 76.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 134.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 253.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 120.T + 2.97e4T^{2} \) |
| 41 | \( 1 - 168.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 47.2T + 7.95e4T^{2} \) |
| 47 | \( 1 - 489.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 763.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 706.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 591.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 854.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 85.5T + 3.57e5T^{2} \) |
| 73 | \( 1 - 857.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 991.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 109.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.21e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.60e3T + 9.12e5T^{2} \) |
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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
−8.002073486961040746861982573524, −7.63336817063564123239809318444, −6.68602530616504448001564801999, −6.22461393608862722080628576264, −5.00431965944382094259706096048, −4.10790465952939065296000729882, −3.33321992885390686687252379913, −2.21090910867065753125175844382, −0.71103670600311487968008117695, 0,
0.71103670600311487968008117695, 2.21090910867065753125175844382, 3.33321992885390686687252379913, 4.10790465952939065296000729882, 5.00431965944382094259706096048, 6.22461393608862722080628576264, 6.68602530616504448001564801999, 7.63336817063564123239809318444, 8.002073486961040746861982573524
