| L(s) = 1 | − 4·2-s + 16·4-s + 25·5-s + 49·7-s − 64·8-s − 100·10-s + 267·11-s − 1.08e3·13-s − 196·14-s + 256·16-s + 513·17-s − 802·19-s + 400·20-s − 1.06e3·22-s + 1.29e3·23-s + 625·25-s + 4.34e3·26-s + 784·28-s − 1.77e3·29-s − 2.58e3·31-s − 1.02e3·32-s − 2.05e3·34-s + 1.22e3·35-s + 1.38e4·37-s + 3.20e3·38-s − 1.60e3·40-s + 1.19e4·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s − 0.353·8-s − 0.316·10-s + 0.665·11-s − 1.78·13-s − 0.267·14-s + 1/4·16-s + 0.430·17-s − 0.509·19-s + 0.223·20-s − 0.470·22-s + 0.508·23-s + 1/5·25-s + 1.26·26-s + 0.188·28-s − 0.392·29-s − 0.482·31-s − 0.176·32-s − 0.304·34-s + 0.169·35-s + 1.66·37-s + 0.360·38-s − 0.158·40-s + 1.10·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.605820458\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.605820458\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{2} T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - p^{2} T \) |
| 7 | \( 1 - p^{2} T \) |
| good | 11 | \( 1 - 267 T + p^{5} T^{2} \) |
| 13 | \( 1 + 1087 T + p^{5} T^{2} \) |
| 17 | \( 1 - 513 T + p^{5} T^{2} \) |
| 19 | \( 1 + 802 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1290 T + p^{5} T^{2} \) |
| 29 | \( 1 + 1779 T + p^{5} T^{2} \) |
| 31 | \( 1 + 2584 T + p^{5} T^{2} \) |
| 37 | \( 1 - 13862 T + p^{5} T^{2} \) |
| 41 | \( 1 - 11904 T + p^{5} T^{2} \) |
| 43 | \( 1 + 598 T + p^{5} T^{2} \) |
| 47 | \( 1 - 17019 T + p^{5} T^{2} \) |
| 53 | \( 1 + 27852 T + p^{5} T^{2} \) |
| 59 | \( 1 + 30912 T + p^{5} T^{2} \) |
| 61 | \( 1 + 1780 T + p^{5} T^{2} \) |
| 67 | \( 1 - 25052 T + p^{5} T^{2} \) |
| 71 | \( 1 - 51984 T + p^{5} T^{2} \) |
| 73 | \( 1 - 47690 T + p^{5} T^{2} \) |
| 79 | \( 1 + 102121 T + p^{5} T^{2} \) |
| 83 | \( 1 - 83676 T + p^{5} T^{2} \) |
| 89 | \( 1 - 32400 T + p^{5} T^{2} \) |
| 97 | \( 1 + 148645 T + p^{5} T^{2} \) |
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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
−9.543521516468268847545391241868, −9.279112150068733273358364179593, −8.008739699851563389654838641839, −7.33983258725116940493253962447, −6.38818975600404573542667400059, −5.35049894994401969123497645093, −4.30475509881734607412857852035, −2.79935584785366393096708195950, −1.87945091548950276710804101626, −0.67540366921610920140395143685,
0.67540366921610920140395143685, 1.87945091548950276710804101626, 2.79935584785366393096708195950, 4.30475509881734607412857852035, 5.35049894994401969123497645093, 6.38818975600404573542667400059, 7.33983258725116940493253962447, 8.008739699851563389654838641839, 9.279112150068733273358364179593, 9.543521516468268847545391241868
