| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s + 9-s + 10-s + 11-s + 12-s + 13-s − 15-s + 16-s + 17-s − 18-s − 8·19-s − 20-s − 22-s − 23-s − 24-s − 4·25-s − 26-s + 27-s − 29-s + 30-s + 10·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.301·11-s + 0.288·12-s + 0.277·13-s − 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 1.83·19-s − 0.223·20-s − 0.213·22-s − 0.208·23-s − 0.204·24-s − 4/5·25-s − 0.196·26-s + 0.192·27-s − 0.185·29-s + 0.182·30-s + 1.79·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54978 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54978 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 7 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.70799647966929, −14.39371651741042, −13.44849914793280, −13.31841506926526, −12.58790431806544, −12.00506329343818, −11.56457893316387, −11.11731897812269, −10.31508479797581, −10.07140593800478, −9.541535977522366, −8.762211795257497, −8.381991940660289, −8.154725077383778, −7.437060184398239, −6.830612968701204, −6.353432083844896, −5.824149319226052, −4.894027149411520, −4.196602030553970, −3.819689303579896, −3.020917670806860, −2.353638965302267, −1.763996600899754, −0.9146213952567641, 0,
0.9146213952567641, 1.763996600899754, 2.353638965302267, 3.020917670806860, 3.819689303579896, 4.196602030553970, 4.894027149411520, 5.824149319226052, 6.353432083844896, 6.830612968701204, 7.437060184398239, 8.154725077383778, 8.381991940660289, 8.762211795257497, 9.541535977522366, 10.07140593800478, 10.31508479797581, 11.11731897812269, 11.56457893316387, 12.00506329343818, 12.58790431806544, 13.31841506926526, 13.44849914793280, 14.39371651741042, 14.70799647966929
