| L(s) = 1 | − 8·2-s + 3-s + 32·4-s + 25·5-s − 8·6-s + 49·7-s − 242·9-s − 200·10-s − 453·11-s + 32·12-s − 969·13-s − 392·14-s + 25·15-s − 1.02e3·16-s + 1.63e3·17-s + 1.93e3·18-s − 1.55e3·19-s + 800·20-s + 49·21-s + 3.62e3·22-s − 1.65e3·23-s + 625·25-s + 7.75e3·26-s − 485·27-s + 1.56e3·28-s − 4.98e3·29-s − 200·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.0641·3-s + 4-s + 0.447·5-s − 0.0907·6-s + 0.377·7-s − 0.995·9-s − 0.632·10-s − 1.12·11-s + 0.0641·12-s − 1.59·13-s − 0.534·14-s + 0.0286·15-s − 16-s + 1.37·17-s + 1.40·18-s − 0.985·19-s + 0.447·20-s + 0.0242·21-s + 1.59·22-s − 0.651·23-s + 1/5·25-s + 2.24·26-s − 0.128·27-s + 0.377·28-s − 1.10·29-s − 0.0405·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - p^{2} T \) |
| 7 | \( 1 - p^{2} T \) |
| good | 2 | \( 1 + p^{3} T + p^{5} T^{2} \) |
| 3 | \( 1 - T + p^{5} T^{2} \) |
| 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
−15.07668629658918242530492204533, −13.87874828629611463272316414379, −12.19534746224870529542228070542, −10.68326794718356656115578889327, −9.799569404919037975018043906064, −8.457587789775511144374168805823, −7.45126682656621643384011516089, −5.36078758673093514145772144574, −2.28257279434830655435142918935, 0,
2.28257279434830655435142918935, 5.36078758673093514145772144574, 7.45126682656621643384011516089, 8.457587789775511144374168805823, 9.799569404919037975018043906064, 10.68326794718356656115578889327, 12.19534746224870529542228070542, 13.87874828629611463272316414379, 15.07668629658918242530492204533
