| L(s) = 1 | + 3.96·2-s − 6.71·3-s + 7.70·4-s − 26.6·6-s + 25.8·7-s − 1.18·8-s + 18.1·9-s − 8.13·11-s − 51.7·12-s + 4.56·13-s + 102.·14-s − 66.2·16-s + 62.5·17-s + 71.7·18-s − 19·19-s − 173.·21-s − 32.2·22-s + 52.7·23-s + 7.93·24-s + 18.0·26-s + 59.7·27-s + 198.·28-s + 171.·29-s + 168.·31-s − 253.·32-s + 54.6·33-s + 247.·34-s + ⋯ |
| L(s) = 1 | + 1.40·2-s − 1.29·3-s + 0.962·4-s − 1.81·6-s + 1.39·7-s − 0.0521·8-s + 0.670·9-s − 0.222·11-s − 1.24·12-s + 0.0974·13-s + 1.95·14-s − 1.03·16-s + 0.892·17-s + 0.939·18-s − 0.229·19-s − 1.80·21-s − 0.312·22-s + 0.478·23-s + 0.0674·24-s + 0.136·26-s + 0.425·27-s + 1.34·28-s + 1.09·29-s + 0.977·31-s − 1.39·32-s + 0.288·33-s + 1.25·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.951994240\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.951994240\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 + 19T \) |
| good | 2 | \( 1 - 3.96T + 8T^{2} \) |
| 3 | \( 1 + 6.71T + 27T^{2} \) |
| 7 | \( 1 - 25.8T + 343T^{2} \) |
| 11 | \( 1 + 8.13T + 1.33e3T^{2} \) |
| 13 | \( 1 - 4.56T + 2.19e3T^{2} \) |
| 17 | \( 1 - 62.5T + 4.91e3T^{2} \) |
| 23 | \( 1 - 52.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 171.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 168.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 147.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 308.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 448.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 113.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 155.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 182.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 404.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 106.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 472.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 843.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 591.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 290.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 964.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 219.T + 9.12e5T^{2} \) |
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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
−11.14257607317209084995767334607, −10.14427045917877166297628027787, −8.659694557468588902537564026628, −7.57002309337987325001596970058, −6.38062699982005776042183722939, −5.63100643969690231456477576396, −4.91283871887317861925698601151, −4.25640372100472110897810053270, −2.68830346625041858979088412528, −0.981616591833813509542561010687,
0.981616591833813509542561010687, 2.68830346625041858979088412528, 4.25640372100472110897810053270, 4.91283871887317861925698601151, 5.63100643969690231456477576396, 6.38062699982005776042183722939, 7.57002309337987325001596970058, 8.659694557468588902537564026628, 10.14427045917877166297628027787, 11.14257607317209084995767334607
