| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 7-s − 8-s + 9-s − 11-s + 12-s − 4·13-s + 14-s + 16-s − 3·17-s − 18-s + 5·19-s − 21-s + 22-s − 3·23-s − 24-s − 5·25-s + 4·26-s + 27-s − 28-s − 6·29-s − 4·31-s − 32-s − 33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s − 1.10·13-s + 0.267·14-s + 1/4·16-s − 0.727·17-s − 0.235·18-s + 1.14·19-s − 0.218·21-s + 0.213·22-s − 0.625·23-s − 0.204·24-s − 25-s + 0.784·26-s + 0.192·27-s − 0.188·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s − 0.174·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4905742033\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4905742033\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 37 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.92953614135721, −13.21759900834373, −13.02068825116240, −12.35085460769281, −11.86128745586941, −11.27025008521577, −10.97410773073167, −10.07466213006440, −9.686127727789430, −9.608780713574775, −8.946135069915522, −8.212071197632684, −7.949858465938982, −7.284036321835078, −7.026195742198606, −6.331732103412631, −5.627993179427529, −5.177014808148603, −4.428493118704130, −3.700797605624900, −3.251184928181681, −2.451081186955586, −2.077119816203534, −1.350830259137053, −0.2332949255622910,
0.2332949255622910, 1.350830259137053, 2.077119816203534, 2.451081186955586, 3.251184928181681, 3.700797605624900, 4.428493118704130, 5.177014808148603, 5.627993179427529, 6.331732103412631, 7.026195742198606, 7.284036321835078, 7.949858465938982, 8.212071197632684, 8.946135069915522, 9.608780713574775, 9.686127727789430, 10.07466213006440, 10.97410773073167, 11.27025008521577, 11.86128745586941, 12.35085460769281, 13.02068825116240, 13.21759900834373, 13.92953614135721
