| L(s) = 1 | − 2·2-s − 8.92·3-s + 4·4-s + 17.8·6-s − 23.5·7-s − 8·8-s + 52.7·9-s + 1.63·11-s − 35.7·12-s + 88.6·13-s + 47.1·14-s + 16·16-s + 7.31·17-s − 105.·18-s − 6.53·19-s + 210.·21-s − 3.27·22-s − 23·23-s + 71.4·24-s − 177.·26-s − 229.·27-s − 94.2·28-s + 119.·29-s − 156.·31-s − 32·32-s − 14.6·33-s − 14.6·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.71·3-s + 0.5·4-s + 1.21·6-s − 1.27·7-s − 0.353·8-s + 1.95·9-s + 0.0448·11-s − 0.859·12-s + 1.89·13-s + 0.899·14-s + 0.250·16-s + 0.104·17-s − 1.38·18-s − 0.0789·19-s + 2.18·21-s − 0.0316·22-s − 0.208·23-s + 0.607·24-s − 1.33·26-s − 1.63·27-s − 0.635·28-s + 0.763·29-s − 0.906·31-s − 0.176·32-s − 0.0770·33-s − 0.0737·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5282234058\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5282234058\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 5 | \( 1 \) |
| 23 | \( 1 + 23T \) |
| good | 3 | \( 1 + 8.92T + 27T^{2} \) |
| 7 | \( 1 + 23.5T + 343T^{2} \) |
| 11 | \( 1 - 1.63T + 1.33e3T^{2} \) |
| 13 | \( 1 - 88.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 7.31T + 4.91e3T^{2} \) |
| 19 | \( 1 + 6.53T + 6.85e3T^{2} \) |
| 29 | \( 1 - 119.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 156.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 293.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 74.3T + 6.89e4T^{2} \) |
| 43 | \( 1 + 468.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 393.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 233.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 766.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 178.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 246.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 650.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 695.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 660.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.32e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 824.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 383.T + 9.12e5T^{2} \) |
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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
−9.587623495418559958413616821986, −8.775045054411185868314819518875, −7.64158450828521431104399941240, −6.55360044949997774916667933792, −6.27683672093954279489346229506, −5.56439446468880103450287802375, −4.25140047878725334557975372134, −3.20767236746668249780698959934, −1.44180770022479117562019226101, −0.49313549747766789060037310329,
0.49313549747766789060037310329, 1.44180770022479117562019226101, 3.20767236746668249780698959934, 4.25140047878725334557975372134, 5.56439446468880103450287802375, 6.27683672093954279489346229506, 6.55360044949997774916667933792, 7.64158450828521431104399941240, 8.775045054411185868314819518875, 9.587623495418559958413616821986
