| L(s) = 1 | + 21.9·2-s − 34.0·3-s + 353.·4-s + 19.5·5-s − 747.·6-s + 1.14e3·7-s + 4.95e3·8-s − 1.02e3·9-s + 427.·10-s − 509.·11-s − 1.20e4·12-s + 6.51e3·13-s + 2.50e4·14-s − 664.·15-s + 6.34e4·16-s − 1.21e3·17-s − 2.25e4·18-s − 1.22e4·19-s + 6.89e3·20-s − 3.88e4·21-s − 1.11e4·22-s − 8.93e4·23-s − 1.68e5·24-s − 7.77e4·25-s + 1.42e5·26-s + 1.09e5·27-s + 4.03e5·28-s + ⋯ |
| L(s) = 1 | + 1.93·2-s − 0.728·3-s + 2.76·4-s + 0.0697·5-s − 1.41·6-s + 1.25·7-s + 3.42·8-s − 0.469·9-s + 0.135·10-s − 0.115·11-s − 2.01·12-s + 0.821·13-s + 2.43·14-s − 0.0508·15-s + 3.87·16-s − 0.0599·17-s − 0.911·18-s − 0.410·19-s + 0.192·20-s − 0.915·21-s − 0.223·22-s − 1.53·23-s − 2.49·24-s − 0.995·25-s + 1.59·26-s + 1.07·27-s + 3.47·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(4.885088500\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.885088500\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + 5.06e4T \) |
| good | 2 | \( 1 - 21.9T + 128T^{2} \) |
| 3 | \( 1 + 34.0T + 2.18e3T^{2} \) |
| 5 | \( 1 - 19.5T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.14e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 509.T + 1.94e7T^{2} \) |
| 13 | \( 1 - 6.51e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.21e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.22e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 8.93e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 6.71e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.46e5T + 2.75e10T^{2} \) |
| 41 | \( 1 - 8.66e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 8.80e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.07e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.41e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.10e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.47e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 9.44e4T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.43e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 4.27e5T + 1.10e13T^{2} \) |
| 79 | \( 1 - 6.99e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.43e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 4.98e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.39e7T + 8.07e13T^{2} \) |
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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
−14.53762369598667636981984175782, −13.79383286497831183594497748272, −12.45204279421622333817743946949, −11.42949639303173181996128206211, −10.84885562428666728788459944375, −7.86950545077561886746325623948, −6.16377601391833094247742134080, −5.30168201768625878042774301102, −3.96475093978772444334821328697, −1.94400106817590540922421432321,
1.94400106817590540922421432321, 3.96475093978772444334821328697, 5.30168201768625878042774301102, 6.16377601391833094247742134080, 7.86950545077561886746325623948, 10.84885562428666728788459944375, 11.42949639303173181996128206211, 12.45204279421622333817743946949, 13.79383286497831183594497748272, 14.53762369598667636981984175782
