| L(s) = 1 | − 3.57·2-s + 4.77·4-s + 14.0·5-s + 0.770·7-s + 11.5·8-s − 50.3·10-s + 34.6·11-s − 15.6·13-s − 2.75·14-s − 79.4·16-s − 0.878·17-s − 23.7·19-s + 67.2·20-s − 123.·22-s − 23·23-s + 73.6·25-s + 55.8·26-s + 3.67·28-s + 208.·29-s + 28.7·31-s + 191.·32-s + 3.14·34-s + 10.8·35-s + 378.·37-s + 84.8·38-s + 162.·40-s − 85.2·41-s + ⋯ |
| L(s) = 1 | − 1.26·2-s + 0.596·4-s + 1.26·5-s + 0.0415·7-s + 0.510·8-s − 1.59·10-s + 0.949·11-s − 0.333·13-s − 0.0525·14-s − 1.24·16-s − 0.0125·17-s − 0.286·19-s + 0.751·20-s − 1.20·22-s − 0.208·23-s + 0.589·25-s + 0.421·26-s + 0.0247·28-s + 1.33·29-s + 0.166·31-s + 1.05·32-s + 0.0158·34-s + 0.0524·35-s + 1.68·37-s + 0.362·38-s + 0.643·40-s − 0.324·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.153529185\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.153529185\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + 23T \) |
| good | 2 | \( 1 + 3.57T + 8T^{2} \) |
| 5 | \( 1 - 14.0T + 125T^{2} \) |
| 7 | \( 1 - 0.770T + 343T^{2} \) |
| 11 | \( 1 - 34.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 15.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 0.878T + 4.91e3T^{2} \) |
| 19 | \( 1 + 23.7T + 6.85e3T^{2} \) |
| 29 | \( 1 - 208.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 28.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 378.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 85.2T + 6.89e4T^{2} \) |
| 43 | \( 1 - 405.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 302.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 49.6T + 1.48e5T^{2} \) |
| 59 | \( 1 - 337.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 439.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 324.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 88.8T + 3.57e5T^{2} \) |
| 73 | \( 1 - 224.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.08e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 463.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.29e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 195.T + 9.12e5T^{2} \) |
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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
−11.68048434666792766345253463111, −10.56559308314759499569603336804, −9.778148034800161567456205761447, −9.180936234345484723334048166480, −8.205375619850013072385626404791, −6.95250112815833210794678515158, −5.95917566277379626665297029651, −4.45290685670348296308001579275, −2.32512030423992637989540071260, −1.05289156654566722191420736601,
1.05289156654566722191420736601, 2.32512030423992637989540071260, 4.45290685670348296308001579275, 5.95917566277379626665297029651, 6.95250112815833210794678515158, 8.205375619850013072385626404791, 9.180936234345484723334048166480, 9.778148034800161567456205761447, 10.56559308314759499569603336804, 11.68048434666792766345253463111
