| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 4·7-s + 8-s + 9-s + 11-s − 12-s − 13-s + 4·14-s + 16-s − 17-s + 18-s − 19-s − 4·21-s + 22-s + 3·23-s − 24-s − 26-s − 27-s + 4·28-s − 6·29-s − 7·31-s + 32-s − 33-s − 34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.301·11-s − 0.288·12-s − 0.277·13-s + 1.06·14-s + 1/4·16-s − 0.242·17-s + 0.235·18-s − 0.229·19-s − 0.872·21-s + 0.213·22-s + 0.625·23-s − 0.204·24-s − 0.196·26-s − 0.192·27-s + 0.755·28-s − 1.11·29-s − 1.25·31-s + 0.176·32-s − 0.174·33-s − 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 364650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 364650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−12.78078368103721, −12.15654955407421, −11.77190604279316, −11.44077078362884, −11.14244752605083, −10.61912546810579, −10.26235430999212, −9.616328345353731, −9.095221871014291, −8.469471307715393, −8.250270846992549, −7.426563734950553, −7.192347390816057, −6.840691639930005, −6.032365085308161, −5.638220090227187, −5.247224509512390, −4.804309344363221, −4.296319357981585, −3.927417882138463, −3.273440320466557, −2.560776421817318, −1.882696382574664, −1.616404614626349, −0.8917575823436549, 0,
0.8917575823436549, 1.616404614626349, 1.882696382574664, 2.560776421817318, 3.273440320466557, 3.927417882138463, 4.296319357981585, 4.804309344363221, 5.247224509512390, 5.638220090227187, 6.032365085308161, 6.840691639930005, 7.192347390816057, 7.426563734950553, 8.250270846992549, 8.469471307715393, 9.095221871014291, 9.616328345353731, 10.26235430999212, 10.61912546810579, 11.14244752605083, 11.44077078362884, 11.77190604279316, 12.15654955407421, 12.78078368103721
