| L(s) = 1 | − 1.86e3·2-s + 5.31e5·3-s − 3.00e7·4-s + 6.53e8·5-s − 9.90e8·6-s − 4.71e10·7-s + 1.18e11·8-s + 2.82e11·9-s − 1.21e12·10-s + 1.60e13·11-s − 1.59e13·12-s + 1.23e14·13-s + 8.79e13·14-s + 3.47e14·15-s + 7.88e14·16-s + 1.37e15·17-s − 5.26e14·18-s + 2.03e14·19-s − 1.96e16·20-s − 2.50e16·21-s − 2.99e16·22-s + 1.96e14·23-s + 6.30e16·24-s + 1.28e17·25-s − 2.29e17·26-s + 1.50e17·27-s + 1.41e18·28-s + ⋯ |
| L(s) = 1 | − 0.321·2-s + 0.577·3-s − 0.896·4-s + 1.19·5-s − 0.185·6-s − 1.28·7-s + 0.610·8-s + 0.333·9-s − 0.385·10-s + 1.54·11-s − 0.517·12-s + 1.46·13-s + 0.414·14-s + 0.690·15-s + 0.700·16-s + 0.574·17-s − 0.107·18-s + 0.0211·19-s − 1.07·20-s − 0.743·21-s − 0.496·22-s + 0.00186·23-s + 0.352·24-s + 0.431·25-s − 0.472·26-s + 0.192·27-s + 1.15·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(13)\) |
\(\approx\) |
\(1.837824689\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.837824689\) |
| \(L(\frac{27}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 5.31e5T \) |
| good | 2 | \( 1 + 1.86e3T + 3.35e7T^{2} \) |
| 5 | \( 1 - 6.53e8T + 2.98e17T^{2} \) |
| 7 | \( 1 + 4.71e10T + 1.34e21T^{2} \) |
| 11 | \( 1 - 1.60e13T + 1.08e26T^{2} \) |
| 13 | \( 1 - 1.23e14T + 7.05e27T^{2} \) |
| 17 | \( 1 - 1.37e15T + 5.77e30T^{2} \) |
| 19 | \( 1 - 2.03e14T + 9.30e31T^{2} \) |
| 23 | \( 1 - 1.96e14T + 1.10e34T^{2} \) |
| 29 | \( 1 - 1.83e18T + 3.63e36T^{2} \) |
| 31 | \( 1 + 4.66e18T + 1.92e37T^{2} \) |
| 37 | \( 1 - 4.09e19T + 1.60e39T^{2} \) |
| 41 | \( 1 + 5.64e19T + 2.08e40T^{2} \) |
| 43 | \( 1 - 1.41e20T + 6.86e40T^{2} \) |
| 47 | \( 1 + 1.37e21T + 6.34e41T^{2} \) |
| 53 | \( 1 - 4.94e21T + 1.27e43T^{2} \) |
| 59 | \( 1 - 3.33e21T + 1.86e44T^{2} \) |
| 61 | \( 1 + 2.98e22T + 4.29e44T^{2} \) |
| 67 | \( 1 - 3.37e22T + 4.48e45T^{2} \) |
| 71 | \( 1 - 1.69e23T + 1.91e46T^{2} \) |
| 73 | \( 1 + 2.16e23T + 3.82e46T^{2} \) |
| 79 | \( 1 - 4.45e23T + 2.75e47T^{2} \) |
| 83 | \( 1 + 1.39e24T + 9.48e47T^{2} \) |
| 89 | \( 1 - 1.32e24T + 5.42e48T^{2} \) |
| 97 | \( 1 + 5.44e24T + 4.66e49T^{2} \) |
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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
−19.62322637099302407945729769988, −18.22844428601110486774267659147, −16.63508741171461474813101967794, −14.14382892795298650748192827213, −13.09064735365258600672812454441, −9.858869994183767283188236942193, −8.934225425099068132186186830283, −6.21897098066779345805889747051, −3.62901817443332944087195956465, −1.23630959236839145675932756156,
1.23630959236839145675932756156, 3.62901817443332944087195956465, 6.21897098066779345805889747051, 8.934225425099068132186186830283, 9.858869994183767283188236942193, 13.09064735365258600672812454441, 14.14382892795298650748192827213, 16.63508741171461474813101967794, 18.22844428601110486774267659147, 19.62322637099302407945729769988
