| L(s) = 1 | − 6.96e3·2-s − 5.31e5·3-s + 1.50e7·4-s + 4.41e8·5-s + 3.70e9·6-s + 2.02e10·7-s + 1.29e11·8-s + 2.82e11·9-s − 3.07e12·10-s − 1.57e13·11-s − 7.97e12·12-s + 4.98e12·13-s − 1.40e14·14-s − 2.34e14·15-s − 1.40e15·16-s − 2.34e15·17-s − 1.96e15·18-s + 1.90e16·19-s + 6.62e15·20-s − 1.07e16·21-s + 1.10e17·22-s − 6.78e16·23-s − 6.86e16·24-s − 1.03e17·25-s − 3.47e16·26-s − 1.50e17·27-s + 3.03e17·28-s + ⋯ |
| L(s) = 1 | − 1.20·2-s − 0.577·3-s + 0.447·4-s + 0.808·5-s + 0.694·6-s + 0.552·7-s + 0.664·8-s + 0.333·9-s − 0.972·10-s − 1.51·11-s − 0.258·12-s + 0.0593·13-s − 0.664·14-s − 0.466·15-s − 1.24·16-s − 0.976·17-s − 0.401·18-s + 1.97·19-s + 0.361·20-s − 0.318·21-s + 1.82·22-s − 0.645·23-s − 0.383·24-s − 0.346·25-s − 0.0714·26-s − 0.192·27-s + 0.247·28-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
| \(L(13)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{27}{2})\) |
|
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 | \( 1 + 6.96e3T + 3.35e7T^{2} \) |
| 5 | \( 1 - 4.41e8T + 2.98e17T^{2} \) |
| 7 | \( 1 - 2.02e10T + 1.34e21T^{2} \) |
| 11 | \( 1 + 1.57e13T + 1.08e26T^{2} \) |
| 13 | \( 1 - 4.98e12T + 7.05e27T^{2} \) |
| 17 | \( 1 + 2.34e15T + 5.77e30T^{2} \) |
| 19 | \( 1 - 1.90e16T + 9.30e31T^{2} \) |
| 23 | \( 1 + 6.78e16T + 1.10e34T^{2} \) |
| 29 | \( 1 + 2.11e18T + 3.63e36T^{2} \) |
| 31 | \( 1 + 6.62e18T + 1.92e37T^{2} \) |
| 37 | \( 1 - 1.02e18T + 1.60e39T^{2} \) |
| 41 | \( 1 + 1.74e20T + 2.08e40T^{2} \) |
| 43 | \( 1 - 3.26e20T + 6.86e40T^{2} \) |
| 47 | \( 1 + 4.36e20T + 6.34e41T^{2} \) |
| 53 | \( 1 + 4.14e21T + 1.27e43T^{2} \) |
| 59 | \( 1 - 1.51e21T + 1.86e44T^{2} \) |
| 61 | \( 1 - 4.63e21T + 4.29e44T^{2} \) |
| 67 | \( 1 + 1.21e23T + 4.48e45T^{2} \) |
| 71 | \( 1 - 1.87e23T + 1.91e46T^{2} \) |
| 73 | \( 1 + 4.86e22T + 3.82e46T^{2} \) |
| 79 | \( 1 - 4.43e23T + 2.75e47T^{2} \) |
| 83 | \( 1 + 1.52e23T + 9.48e47T^{2} \) |
| 89 | \( 1 - 1.97e24T + 5.42e48T^{2} \) |
| 97 | \( 1 + 3.28e24T + 4.66e49T^{2} \) |
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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
−18.31800289609088332910017940263, −17.72926106219290764723810215068, −16.09161489194395351973017042675, −13.47283324448360023906937134211, −10.97677971301561767639065507078, −9.590651031119803877311337379530, −7.65801652679983552938765927446, −5.28494678642492329572697611963, −1.78130573606761996021613258231, 0,
1.78130573606761996021613258231, 5.28494678642492329572697611963, 7.65801652679983552938765927446, 9.590651031119803877311337379530, 10.97677971301561767639065507078, 13.47283324448360023906937134211, 16.09161489194395351973017042675, 17.72926106219290764723810215068, 18.31800289609088332910017940263
