| L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s + 7.60·5-s + 6·6-s − 24.7·7-s + 8·8-s + 9·9-s + 15.2·10-s − 10.6·11-s + 12·12-s + 73.6·13-s − 49.4·14-s + 22.8·15-s + 16·16-s + 65.3·17-s + 18·18-s + 161.·19-s + 30.4·20-s − 74.1·21-s − 21.2·22-s + 81.2·23-s + 24·24-s − 67.1·25-s + 147.·26-s + 27·27-s − 98.8·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.680·5-s + 0.408·6-s − 1.33·7-s + 0.353·8-s + 0.333·9-s + 0.481·10-s − 0.291·11-s + 0.288·12-s + 1.57·13-s − 0.943·14-s + 0.392·15-s + 0.250·16-s + 0.931·17-s + 0.235·18-s + 1.94·19-s + 0.340·20-s − 0.770·21-s − 0.206·22-s + 0.736·23-s + 0.204·24-s − 0.537·25-s + 1.11·26-s + 0.192·27-s − 0.666·28-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
| \(L(2)\) |
\(\approx\) |
\(3.897492365\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.897492365\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 7.60T + 125T^{2} \) |
| 7 | \( 1 + 24.7T + 343T^{2} \) |
| 11 | \( 1 + 10.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 73.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 65.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 161.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 81.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 276.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 252.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 149.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 195.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 322.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 536.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 155.T + 1.48e5T^{2} \) |
| 61 | \( 1 + 364.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 26.9T + 3.00e5T^{2} \) |
| 71 | \( 1 + 106.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.14e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 528.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 768.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 804.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.22e3T + 9.12e5T^{2} \) |
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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
−11.05059457585440339563139893738, −9.963688444988701120388190518893, −9.422242329409432865701012224875, −8.200209116282103551246495016966, −7.01382623659639869451709119307, −6.12016704001864839296203764860, −5.24041597094398866342591030610, −3.52223561796638240572051076763, −3.05437799873637390157953675267, −1.33536586434539823265113553730,
1.33536586434539823265113553730, 3.05437799873637390157953675267, 3.52223561796638240572051076763, 5.24041597094398866342591030610, 6.12016704001864839296203764860, 7.01382623659639869451709119307, 8.200209116282103551246495016966, 9.422242329409432865701012224875, 9.963688444988701120388190518893, 11.05059457585440339563139893738
