| L(s) = 1 | − 4.20·2-s − 3·3-s + 9.69·4-s − 5.61·5-s + 12.6·6-s + 32.2·7-s − 7.14·8-s + 9·9-s + 23.6·10-s − 11·11-s − 29.0·12-s + 55.5·13-s − 135.·14-s + 16.8·15-s − 47.5·16-s − 77.6·17-s − 37.8·18-s − 129.·19-s − 54.4·20-s − 96.6·21-s + 46.2·22-s − 105.·23-s + 21.4·24-s − 93.4·25-s − 233.·26-s − 27·27-s + 312.·28-s + ⋯ |
| L(s) = 1 | − 1.48·2-s − 0.577·3-s + 1.21·4-s − 0.502·5-s + 0.858·6-s + 1.73·7-s − 0.315·8-s + 0.333·9-s + 0.746·10-s − 0.301·11-s − 0.699·12-s + 1.18·13-s − 2.58·14-s + 0.289·15-s − 0.742·16-s − 1.10·17-s − 0.495·18-s − 1.55·19-s − 0.608·20-s − 1.00·21-s + 0.448·22-s − 0.952·23-s + 0.182·24-s − 0.747·25-s − 1.76·26-s − 0.192·27-s + 2.10·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6455559283\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6455559283\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 11 | \( 1 + 11T \) |
| 61 | \( 1 + 61T \) |
| good | 2 | \( 1 + 4.20T + 8T^{2} \) |
| 5 | \( 1 + 5.61T + 125T^{2} \) |
| 7 | \( 1 - 32.2T + 343T^{2} \) |
| 13 | \( 1 - 55.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 77.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 129.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 105.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 19.9T + 2.43e4T^{2} \) |
| 31 | \( 1 - 101.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 171.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 353.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 203.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 209.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 383.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 147.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 135.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 531.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 582.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.24e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 848.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 559.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 227.T + 9.12e5T^{2} \) |
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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
−8.639127009842384755260808203930, −8.033989529349443835907411935112, −7.72944199787078513980824336870, −6.59029978652139463843895033759, −5.87318311560345405355479593094, −4.51155142319387800045981204838, −4.22283611788941674385213654853, −2.24014598487756767683597037129, −1.55907776517434628503822196884, −0.49904087863517984471927281927,
0.49904087863517984471927281927, 1.55907776517434628503822196884, 2.24014598487756767683597037129, 4.22283611788941674385213654853, 4.51155142319387800045981204838, 5.87318311560345405355479593094, 6.59029978652139463843895033759, 7.72944199787078513980824336870, 8.033989529349443835907411935112, 8.639127009842384755260808203930
