| L(s) = 1 | − 4.09e3·2-s + 1.67e7·4-s − 8.78e8·5-s + 3.00e10·7-s − 6.87e10·8-s + 3.60e12·10-s − 5.73e12·11-s − 1.07e13·13-s − 1.23e14·14-s + 2.81e14·16-s − 2.97e15·17-s − 5.42e15·19-s − 1.47e16·20-s + 2.34e16·22-s + 1.04e17·23-s + 4.74e17·25-s + 4.39e16·26-s + 5.03e17·28-s − 3.09e18·29-s − 4.26e18·31-s − 1.15e18·32-s + 1.21e19·34-s − 2.64e19·35-s − 4.51e19·37-s + 2.22e19·38-s + 6.04e19·40-s + 7.56e19·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.61·5-s + 0.820·7-s − 0.353·8-s + 1.13·10-s − 0.551·11-s − 0.127·13-s − 0.580·14-s + 0.250·16-s − 1.23·17-s − 0.562·19-s − 0.805·20-s + 0.389·22-s + 0.994·23-s + 1.59·25-s + 0.0903·26-s + 0.410·28-s − 1.62·29-s − 0.973·31-s − 0.176·32-s + 0.875·34-s − 1.32·35-s − 1.12·37-s + 0.397·38-s + 0.569·40-s + 0.523·41-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
| \(L(13)\) |
\(\approx\) |
\(0.5652664451\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5652664451\) |
| \(L(\frac{27}{2})\) |
|
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 | \( 1 + 8.78e8T + 2.98e17T^{2} \) |
| 7 | \( 1 - 3.00e10T + 1.34e21T^{2} \) |
| 11 | \( 1 + 5.73e12T + 1.08e26T^{2} \) |
| 13 | \( 1 + 1.07e13T + 7.05e27T^{2} \) |
| 17 | \( 1 + 2.97e15T + 5.77e30T^{2} \) |
| 19 | \( 1 + 5.42e15T + 9.30e31T^{2} \) |
| 23 | \( 1 - 1.04e17T + 1.10e34T^{2} \) |
| 29 | \( 1 + 3.09e18T + 3.63e36T^{2} \) |
| 31 | \( 1 + 4.26e18T + 1.92e37T^{2} \) |
| 37 | \( 1 + 4.51e19T + 1.60e39T^{2} \) |
| 41 | \( 1 - 7.56e19T + 2.08e40T^{2} \) |
| 43 | \( 1 + 1.39e20T + 6.86e40T^{2} \) |
| 47 | \( 1 + 4.67e20T + 6.34e41T^{2} \) |
| 53 | \( 1 - 1.24e21T + 1.27e43T^{2} \) |
| 59 | \( 1 - 1.32e22T + 1.86e44T^{2} \) |
| 61 | \( 1 + 2.36e20T + 4.29e44T^{2} \) |
| 67 | \( 1 + 5.36e22T + 4.48e45T^{2} \) |
| 71 | \( 1 - 2.27e23T + 1.91e46T^{2} \) |
| 73 | \( 1 - 2.78e23T + 3.82e46T^{2} \) |
| 79 | \( 1 - 6.36e23T + 2.75e47T^{2} \) |
| 83 | \( 1 + 1.19e24T + 9.48e47T^{2} \) |
| 89 | \( 1 - 3.22e24T + 5.42e48T^{2} \) |
| 97 | \( 1 - 3.58e24T + 4.66e49T^{2} \) |
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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
−12.83013168040266748110854379406, −11.43309906837019624662161005762, −10.86376683054453447314147730428, −8.911294944649996145230102612745, −7.946161488155440123082930209399, −7.01820866281378949982093163759, −4.94884701779025338058325453859, −3.63730450553272444843802071264, −2.01854088583459914493392596188, −0.42036090688098676328035739724,
0.42036090688098676328035739724, 2.01854088583459914493392596188, 3.63730450553272444843802071264, 4.94884701779025338058325453859, 7.01820866281378949982093163759, 7.946161488155440123082930209399, 8.911294944649996145230102612745, 10.86376683054453447314147730428, 11.43309906837019624662161005762, 12.83013168040266748110854379406
