| L(s) = 1 | + 8·2-s + 32·4-s − 25·5-s + 49·7-s − 200·10-s + 453·11-s − 969·13-s + 392·14-s − 1.02e3·16-s − 1.63e3·17-s − 1.55e3·19-s − 800·20-s + 3.62e3·22-s + 1.65e3·23-s + 625·25-s − 7.75e3·26-s + 1.56e3·28-s + 4.98e3·29-s + 1.19e3·31-s − 8.19e3·32-s − 1.30e4·34-s − 1.22e3·35-s − 1.10e4·37-s − 1.24e4·38-s + 1.72e3·41-s − 1.08e4·43-s + 1.44e4·44-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 0.447·5-s + 0.377·7-s − 0.632·10-s + 1.12·11-s − 1.59·13-s + 0.534·14-s − 16-s − 1.37·17-s − 0.985·19-s − 0.447·20-s + 1.59·22-s + 0.651·23-s + 1/5·25-s − 2.24·26-s + 0.377·28-s + 1.10·29-s + 0.222·31-s − 1.41·32-s − 1.94·34-s − 0.169·35-s − 1.32·37-s − 1.39·38-s + 0.160·41-s − 0.891·43-s + 1.12·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 + p^{2} T \) |
| 7 | \( 1 - p^{2} T \) |
| good | 2 | \( 1 - p^{3} T + p^{5} T^{2} \) |
| 11 | \( 1 - 453 T + p^{5} T^{2} \) |
| 13 | \( 1 + 969 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1637 T + p^{5} T^{2} \) |
| 19 | \( 1 + 1550 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1654 T + p^{5} T^{2} \) |
| 29 | \( 1 - 4985 T + p^{5} T^{2} \) |
| 31 | \( 1 - 1192 T + p^{5} T^{2} \) |
| 37 | \( 1 + 11018 T + p^{5} T^{2} \) |
| 41 | \( 1 - 1728 T + p^{5} T^{2} \) |
| 43 | \( 1 + 10814 T + p^{5} T^{2} \) |
| 47 | \( 1 + 26237 T + p^{5} T^{2} \) |
| 53 | \( 1 + 25936 T + p^{5} T^{2} \) |
| 59 | \( 1 - 4580 T + p^{5} T^{2} \) |
| 61 | \( 1 + 12488 T + p^{5} T^{2} \) |
| 67 | \( 1 + 15848 T + p^{5} T^{2} \) |
| 71 | \( 1 + 51792 T + p^{5} T^{2} \) |
| 73 | \( 1 - 4846 T + p^{5} T^{2} \) |
| 79 | \( 1 - 62765 T + p^{5} T^{2} \) |
| 83 | \( 1 - 23644 T + p^{5} T^{2} \) |
| 89 | \( 1 - 147300 T + p^{5} T^{2} \) |
| 97 | \( 1 + 8343 T + p^{5} T^{2} \) |
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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
−10.72762204268833110805514449697, −9.365297930388487405630285878003, −8.432611128998689790895892673645, −6.98997681832332511007365732938, −6.40960342335006387101201120503, −4.85055673527880708862550159975, −4.50470102170429987947047762813, −3.24448811898225928719541408102, −2.01880562975711315218012924910, 0,
2.01880562975711315218012924910, 3.24448811898225928719541408102, 4.50470102170429987947047762813, 4.85055673527880708862550159975, 6.40960342335006387101201120503, 6.98997681832332511007365732938, 8.432611128998689790895892673645, 9.365297930388487405630285878003, 10.72762204268833110805514449697
