| L(s) = 1 | − 9·3-s + 53.3·5-s − 89.8·7-s + 81·9-s − 225.·11-s + 725.·13-s − 480.·15-s + 44.9·17-s − 1.21e3·19-s + 808.·21-s + 529·23-s − 274.·25-s − 729·27-s + 2.59e3·29-s − 585.·31-s + 2.02e3·33-s − 4.79e3·35-s − 3.84e3·37-s − 6.53e3·39-s − 4.29e3·41-s + 2.05e4·43-s + 4.32e3·45-s + 5.22e3·47-s − 8.73e3·49-s − 404.·51-s − 1.57e4·53-s − 1.20e4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.955·5-s − 0.693·7-s + 0.333·9-s − 0.560·11-s + 1.19·13-s − 0.551·15-s + 0.0377·17-s − 0.770·19-s + 0.400·21-s + 0.208·23-s − 0.0878·25-s − 0.192·27-s + 0.572·29-s − 0.109·31-s + 0.323·33-s − 0.662·35-s − 0.462·37-s − 0.687·39-s − 0.399·41-s + 1.69·43-s + 0.318·45-s + 0.344·47-s − 0.519·49-s − 0.0217·51-s − 0.770·53-s − 0.535·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 9T \) |
| 23 | \( 1 - 529T \) |
| good | 5 | \( 1 - 53.3T + 3.12e3T^{2} \) |
| 7 | \( 1 + 89.8T + 1.68e4T^{2} \) |
| 11 | \( 1 + 225.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 725.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 44.9T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.21e3T + 2.47e6T^{2} \) |
| 29 | \( 1 - 2.59e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 585.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 3.84e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 4.29e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.05e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 5.22e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.57e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.37e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.89e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.95e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.52e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.17e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 9.58e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.33e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.09e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 4.42e4T + 8.58e9T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.880579498854941032624757105770, −7.88437722759747433983404753011, −6.74404451631198820586010816399, −6.12562190980901169502400043542, −5.54124178840647782308821832486, −4.44713229408813849367315435412, −3.34517199943005978841056505261, −2.23687389185877669370638754160, −1.17233848934372456916050307208, 0,
1.17233848934372456916050307208, 2.23687389185877669370638754160, 3.34517199943005978841056505261, 4.44713229408813849367315435412, 5.54124178840647782308821832486, 6.12562190980901169502400043542, 6.74404451631198820586010816399, 7.88437722759747433983404753011, 8.880579498854941032624757105770
