| L(s) = 1 | − 8·2-s + 27·3-s + 64·4-s − 465.·5-s − 216·6-s + 1.57e3·7-s − 512·8-s + 729·9-s + 3.72e3·10-s + 6.19e3·11-s + 1.72e3·12-s − 1.40e4·13-s − 1.25e4·14-s − 1.25e4·15-s + 4.09e3·16-s − 2.31e4·17-s − 5.83e3·18-s + 3.19e4·19-s − 2.98e4·20-s + 4.23e4·21-s − 4.95e4·22-s + 1.21e4·23-s − 1.38e4·24-s + 1.38e5·25-s + 1.12e5·26-s + 1.96e4·27-s + 1.00e5·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.66·5-s − 0.408·6-s + 1.73·7-s − 0.353·8-s + 0.333·9-s + 1.17·10-s + 1.40·11-s + 0.288·12-s − 1.77·13-s − 1.22·14-s − 0.962·15-s + 0.250·16-s − 1.14·17-s − 0.235·18-s + 1.07·19-s − 0.833·20-s + 0.998·21-s − 0.992·22-s + 0.208·23-s − 0.204·24-s + 1.77·25-s + 1.25·26-s + 0.192·27-s + 0.865·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.551549443\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.551549443\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 8T \) |
| 3 | \( 1 - 27T \) |
| 23 | \( 1 - 1.21e4T \) |
| good | 5 | \( 1 + 465.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.57e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 6.19e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.40e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.31e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.19e4T + 8.93e8T^{2} \) |
| 29 | \( 1 + 1.05e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.19e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.04e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 8.12e3T + 1.94e11T^{2} \) |
| 43 | \( 1 - 9.55e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 5.77e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 8.71e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.21e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.52e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.86e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.37e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 5.73e5T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.85e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 1.36e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.82e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 9.77e6T + 8.07e13T^{2} \) |
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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
−11.53283280076578276012248112232, −11.17284710307034182276184820152, −9.475940535898519395076677426082, −8.584502691199220239341324314375, −7.61556201803272231801953429752, −7.19221916327566627603552687449, −4.83717270632665592016995588405, −3.88927683010727273945098771262, −2.19831766456289812018293232334, −0.795335927442366299803554041900,
0.795335927442366299803554041900, 2.19831766456289812018293232334, 3.88927683010727273945098771262, 4.83717270632665592016995588405, 7.19221916327566627603552687449, 7.61556201803272231801953429752, 8.584502691199220239341324314375, 9.475940535898519395076677426082, 11.17284710307034182276184820152, 11.53283280076578276012248112232
