| L(s) = 1 | − 6.64e3·2-s + 1.05e7·4-s − 1.29e8·5-s − 4.99e10·7-s + 1.52e11·8-s + 8.60e11·10-s + 2.40e12·11-s − 1.10e14·13-s + 3.31e14·14-s − 1.36e15·16-s − 1.88e15·17-s − 5.16e15·19-s − 1.37e15·20-s − 1.59e16·22-s − 2.07e17·23-s − 2.81e17·25-s + 7.31e17·26-s − 5.29e17·28-s + 3.91e17·29-s + 7.70e18·31-s + 3.97e18·32-s + 1.25e19·34-s + 6.46e18·35-s − 2.99e18·37-s + 3.43e19·38-s − 1.97e19·40-s + 2.45e20·41-s + ⋯ |
| L(s) = 1 | − 1.14·2-s + 0.315·4-s − 0.237·5-s − 1.36·7-s + 0.784·8-s + 0.272·10-s + 0.230·11-s − 1.31·13-s + 1.56·14-s − 1.21·16-s − 0.786·17-s − 0.535·19-s − 0.0749·20-s − 0.264·22-s − 1.97·23-s − 0.943·25-s + 1.50·26-s − 0.430·28-s + 0.205·29-s + 1.75·31-s + 0.610·32-s + 0.901·34-s + 0.323·35-s − 0.0748·37-s + 0.614·38-s − 0.186·40-s + 1.69·41-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
| \(L(13)\) |
\(\approx\) |
\(0.2913031200\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2913031200\) |
| \(L(\frac{27}{2})\) |
|
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 | \( 1 + 6.64e3T + 3.35e7T^{2} \) |
| 5 | \( 1 + 1.29e8T + 2.98e17T^{2} \) |
| 7 | \( 1 + 4.99e10T + 1.34e21T^{2} \) |
| 11 | \( 1 - 2.40e12T + 1.08e26T^{2} \) |
| 13 | \( 1 + 1.10e14T + 7.05e27T^{2} \) |
| 17 | \( 1 + 1.88e15T + 5.77e30T^{2} \) |
| 19 | \( 1 + 5.16e15T + 9.30e31T^{2} \) |
| 23 | \( 1 + 2.07e17T + 1.10e34T^{2} \) |
| 29 | \( 1 - 3.91e17T + 3.63e36T^{2} \) |
| 31 | \( 1 - 7.70e18T + 1.92e37T^{2} \) |
| 37 | \( 1 + 2.99e18T + 1.60e39T^{2} \) |
| 41 | \( 1 - 2.45e20T + 2.08e40T^{2} \) |
| 43 | \( 1 + 2.88e20T + 6.86e40T^{2} \) |
| 47 | \( 1 - 3.01e20T + 6.34e41T^{2} \) |
| 53 | \( 1 - 4.35e21T + 1.27e43T^{2} \) |
| 59 | \( 1 + 6.92e21T + 1.86e44T^{2} \) |
| 61 | \( 1 + 2.89e22T + 4.29e44T^{2} \) |
| 67 | \( 1 + 1.41e22T + 4.48e45T^{2} \) |
| 71 | \( 1 - 6.66e22T + 1.91e46T^{2} \) |
| 73 | \( 1 - 1.33e23T + 3.82e46T^{2} \) |
| 79 | \( 1 - 8.16e23T + 2.75e47T^{2} \) |
| 83 | \( 1 - 7.24e23T + 9.48e47T^{2} \) |
| 89 | \( 1 + 1.55e23T + 5.42e48T^{2} \) |
| 97 | \( 1 + 2.30e24T + 4.66e49T^{2} \) |
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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
−15.63873336097921185543997058778, −13.68107491323425869770770025495, −12.13904401583450031387442993937, −10.23486938707575233241450797526, −9.400709210290287216166940925236, −7.901149392687559199063134880445, −6.47198775740922140209715202616, −4.22706796403521613357784374587, −2.30027280479886149476266860066, −0.36830474762161133595401064889,
0.36830474762161133595401064889, 2.30027280479886149476266860066, 4.22706796403521613357784374587, 6.47198775740922140209715202616, 7.901149392687559199063134880445, 9.400709210290287216166940925236, 10.23486938707575233241450797526, 12.13904401583450031387442993937, 13.68107491323425869770770025495, 15.63873336097921185543997058778
