L(s) = 1 | − 5.33·2-s − 3·3-s + 20.4·4-s + 16.4·5-s + 15.9·6-s − 9.67·7-s − 66.1·8-s + 9·9-s − 87.4·10-s − 27.5·11-s − 61.2·12-s + 51.5·14-s − 49.2·15-s + 189.·16-s + 107.·17-s − 47.9·18-s + 2.24·19-s + 335.·20-s + 29.0·21-s + 147.·22-s + 41.8·23-s + 198.·24-s + 144.·25-s − 27·27-s − 197.·28-s + 61.6·29-s + 262.·30-s + ⋯ |
L(s) = 1 | − 1.88·2-s − 0.577·3-s + 2.55·4-s + 1.46·5-s + 1.08·6-s − 0.522·7-s − 2.92·8-s + 0.333·9-s − 2.76·10-s − 0.756·11-s − 1.47·12-s + 0.984·14-s − 0.847·15-s + 2.95·16-s + 1.53·17-s − 0.628·18-s + 0.0271·19-s + 3.74·20-s + 0.301·21-s + 1.42·22-s + 0.379·23-s + 1.68·24-s + 1.15·25-s − 0.192·27-s − 1.33·28-s + 0.394·29-s + 1.59·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.7776243021\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7776243021\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 3T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 5.33T + 8T^{2} \) |
| 5 | \( 1 - 16.4T + 125T^{2} \) |
| 7 | \( 1 + 9.67T + 343T^{2} \) |
| 11 | \( 1 + 27.5T + 1.33e3T^{2} \) |
| 17 | \( 1 - 107.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 2.24T + 6.85e3T^{2} \) |
| 23 | \( 1 - 41.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 61.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 191.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 98.4T + 5.06e4T^{2} \) |
| 41 | \( 1 - 30.7T + 6.89e4T^{2} \) |
| 43 | \( 1 - 238.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 511.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 492.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 484.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 444.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 190.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 484.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 957.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 375.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 715.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.03e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 65.5T + 9.12e5T^{2} \) |
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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
−10.43704214053857339340243697692, −9.589345830053885513124315965348, −9.086818089036718198599342576820, −7.84749590523423072721259012060, −7.03992908244070721783683507215, −6.03815203136158766908090607279, −5.46299961969421959997042950031, −2.99122005000924061146634408929, −1.86981230770441601724898593627, −0.74850480592895428022114320089,
0.74850480592895428022114320089, 1.86981230770441601724898593627, 2.99122005000924061146634408929, 5.46299961969421959997042950031, 6.03815203136158766908090607279, 7.03992908244070721783683507215, 7.84749590523423072721259012060, 9.086818089036718198599342576820, 9.589345830053885513124315965348, 10.43704214053857339340243697692