| L(s) = 1 | − 8.74e3·2-s + 4.29e7·4-s − 4.98e7·5-s + 5.50e10·7-s − 8.17e10·8-s + 4.35e11·10-s + 1.39e13·11-s + 1.27e14·13-s − 4.81e14·14-s − 7.24e14·16-s − 1.41e15·17-s − 1.02e16·19-s − 2.13e15·20-s − 1.22e17·22-s + 6.75e16·23-s − 2.95e17·25-s − 1.11e18·26-s + 2.36e18·28-s + 3.07e18·29-s + 9.03e17·31-s + 9.08e18·32-s + 1.24e19·34-s − 2.74e18·35-s − 5.36e18·37-s + 8.99e19·38-s + 4.07e18·40-s − 1.40e20·41-s + ⋯ |
| L(s) = 1 | − 1.50·2-s + 1.27·4-s − 0.0912·5-s + 1.50·7-s − 0.420·8-s + 0.137·10-s + 1.34·11-s + 1.51·13-s − 2.26·14-s − 0.643·16-s − 0.590·17-s − 1.06·19-s − 0.116·20-s − 2.02·22-s + 0.643·23-s − 0.991·25-s − 2.29·26-s + 1.92·28-s + 1.61·29-s + 0.205·31-s + 1.39·32-s + 0.891·34-s − 0.137·35-s − 0.134·37-s + 1.61·38-s + 0.0384·40-s − 0.974·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(13)\) |
\(\approx\) |
\(1.280054296\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.280054296\) |
| \(L(\frac{27}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + 8.74e3T + 3.35e7T^{2} \) |
| 5 | \( 1 + 4.98e7T + 2.98e17T^{2} \) |
| 7 | \( 1 - 5.50e10T + 1.34e21T^{2} \) |
| 11 | \( 1 - 1.39e13T + 1.08e26T^{2} \) |
| 13 | \( 1 - 1.27e14T + 7.05e27T^{2} \) |
| 17 | \( 1 + 1.41e15T + 5.77e30T^{2} \) |
| 19 | \( 1 + 1.02e16T + 9.30e31T^{2} \) |
| 23 | \( 1 - 6.75e16T + 1.10e34T^{2} \) |
| 29 | \( 1 - 3.07e18T + 3.63e36T^{2} \) |
| 31 | \( 1 - 9.03e17T + 1.92e37T^{2} \) |
| 37 | \( 1 + 5.36e18T + 1.60e39T^{2} \) |
| 41 | \( 1 + 1.40e20T + 2.08e40T^{2} \) |
| 43 | \( 1 - 1.70e20T + 6.86e40T^{2} \) |
| 47 | \( 1 - 6.32e20T + 6.34e41T^{2} \) |
| 53 | \( 1 + 9.60e20T + 1.27e43T^{2} \) |
| 59 | \( 1 + 1.44e22T + 1.86e44T^{2} \) |
| 61 | \( 1 - 3.87e21T + 4.29e44T^{2} \) |
| 67 | \( 1 - 8.52e22T + 4.48e45T^{2} \) |
| 71 | \( 1 + 7.92e22T + 1.91e46T^{2} \) |
| 73 | \( 1 - 2.67e23T + 3.82e46T^{2} \) |
| 79 | \( 1 + 6.21e23T + 2.75e47T^{2} \) |
| 83 | \( 1 + 3.18e23T + 9.48e47T^{2} \) |
| 89 | \( 1 - 3.79e24T + 5.42e48T^{2} \) |
| 97 | \( 1 - 1.55e24T + 4.66e49T^{2} \) |
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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
−15.53022781831421821523575371594, −13.95533573630117373725609972209, −11.59038117103746022687355880482, −10.71352486519984034920197962172, −8.922667313366115424906628480138, −8.189672218821028325517795210232, −6.55683577547876976356563499475, −4.29545692114996162540600878747, −1.82119116541440768349269492344, −0.945506555640721259234792976530,
0.945506555640721259234792976530, 1.82119116541440768349269492344, 4.29545692114996162540600878747, 6.55683577547876976356563499475, 8.189672218821028325517795210232, 8.922667313366115424906628480138, 10.71352486519984034920197962172, 11.59038117103746022687355880482, 13.95533573630117373725609972209, 15.53022781831421821523575371594
