| L(s) = 1 | + 4.09e3·2-s + 1.67e7·4-s + 5.87e8·5-s − 6.29e10·7-s + 6.87e10·8-s + 2.40e12·10-s − 1.65e13·11-s + 6.25e13·13-s − 2.57e14·14-s + 2.81e14·16-s + 4.12e15·17-s + 1.78e16·19-s + 9.84e15·20-s − 6.77e16·22-s − 5.26e16·23-s + 4.66e16·25-s + 2.56e17·26-s − 1.05e18·28-s + 4.67e15·29-s + 5.16e18·31-s + 1.15e18·32-s + 1.68e19·34-s − 3.69e19·35-s − 1.72e19·37-s + 7.32e19·38-s + 4.03e19·40-s + 1.59e20·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.07·5-s − 1.71·7-s + 0.353·8-s + 0.760·10-s − 1.59·11-s + 0.744·13-s − 1.21·14-s + 0.250·16-s + 1.71·17-s + 1.85·19-s + 0.537·20-s − 1.12·22-s − 0.501·23-s + 0.156·25-s + 0.526·26-s − 0.859·28-s + 0.00245·29-s + 1.17·31-s + 0.176·32-s + 1.21·34-s − 1.84·35-s − 0.429·37-s + 1.31·38-s + 0.380·40-s + 1.10·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(13)\) |
\(\approx\) |
\(3.563120465\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.563120465\) |
| \(L(\frac{27}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4.09e3T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 5.87e8T + 2.98e17T^{2} \) |
| 7 | \( 1 + 6.29e10T + 1.34e21T^{2} \) |
| 11 | \( 1 + 1.65e13T + 1.08e26T^{2} \) |
| 13 | \( 1 - 6.25e13T + 7.05e27T^{2} \) |
| 17 | \( 1 - 4.12e15T + 5.77e30T^{2} \) |
| 19 | \( 1 - 1.78e16T + 9.30e31T^{2} \) |
| 23 | \( 1 + 5.26e16T + 1.10e34T^{2} \) |
| 29 | \( 1 - 4.67e15T + 3.63e36T^{2} \) |
| 31 | \( 1 - 5.16e18T + 1.92e37T^{2} \) |
| 37 | \( 1 + 1.72e19T + 1.60e39T^{2} \) |
| 41 | \( 1 - 1.59e20T + 2.08e40T^{2} \) |
| 43 | \( 1 - 3.44e20T + 6.86e40T^{2} \) |
| 47 | \( 1 + 4.59e20T + 6.34e41T^{2} \) |
| 53 | \( 1 - 4.24e21T + 1.27e43T^{2} \) |
| 59 | \( 1 - 6.73e19T + 1.86e44T^{2} \) |
| 61 | \( 1 + 8.44e21T + 4.29e44T^{2} \) |
| 67 | \( 1 - 5.11e21T + 4.48e45T^{2} \) |
| 71 | \( 1 + 4.17e22T + 1.91e46T^{2} \) |
| 73 | \( 1 + 1.19e22T + 3.82e46T^{2} \) |
| 79 | \( 1 + 1.70e23T + 2.75e47T^{2} \) |
| 83 | \( 1 - 1.85e24T + 9.48e47T^{2} \) |
| 89 | \( 1 - 3.40e23T + 5.42e48T^{2} \) |
| 97 | \( 1 - 4.10e24T + 4.66e49T^{2} \) |
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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
−13.29025538356573260248694355174, −12.24320438906440205358758488022, −10.31725692648059729657034143457, −9.643163398822211696697852209018, −7.58596706035431859201240617467, −6.05470537866162515811727376052, −5.45172572036970947509797552040, −3.40539839003672201218571436173, −2.61672789267756907276522717569, −0.891863108533529031959436572112,
0.891863108533529031959436572112, 2.61672789267756907276522717569, 3.40539839003672201218571436173, 5.45172572036970947509797552040, 6.05470537866162515811727376052, 7.58596706035431859201240617467, 9.643163398822211696697852209018, 10.31725692648059729657034143457, 12.24320438906440205358758488022, 13.29025538356573260248694355174
