| L(s) = 1 | − 2·2-s + 4·4-s + 17·5-s + 7·7-s − 8·8-s − 34·10-s + 11·11-s − 21·13-s − 14·14-s + 16·16-s + 104·17-s − 161·19-s + 68·20-s − 22·22-s − 194·23-s + 164·25-s + 42·26-s + 28·28-s − 9·29-s − 180·31-s − 32·32-s − 208·34-s + 119·35-s − 363·37-s + 322·38-s − 136·40-s + 108·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.52·5-s + 0.377·7-s − 0.353·8-s − 1.07·10-s + 0.301·11-s − 0.448·13-s − 0.267·14-s + 1/4·16-s + 1.48·17-s − 1.94·19-s + 0.760·20-s − 0.213·22-s − 1.75·23-s + 1.31·25-s + 0.316·26-s + 0.188·28-s − 0.0576·29-s − 1.04·31-s − 0.176·32-s − 1.04·34-s + 0.574·35-s − 1.61·37-s + 1.37·38-s − 0.537·40-s + 0.411·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - p T \) |
| 11 | \( 1 - p T \) |
| good | 5 | \( 1 - 17 T + p^{3} T^{2} \) |
| 13 | \( 1 + 21 T + p^{3} T^{2} \) |
| 17 | \( 1 - 104 T + p^{3} T^{2} \) |
| 19 | \( 1 + 161 T + p^{3} T^{2} \) |
| 23 | \( 1 + 194 T + p^{3} T^{2} \) |
| 29 | \( 1 + 9 T + p^{3} T^{2} \) |
| 31 | \( 1 + 180 T + p^{3} T^{2} \) |
| 37 | \( 1 + 363 T + p^{3} T^{2} \) |
| 41 | \( 1 - 108 T + p^{3} T^{2} \) |
| 43 | \( 1 + 386 T + p^{3} T^{2} \) |
| 47 | \( 1 + 333 T + p^{3} T^{2} \) |
| 53 | \( 1 - 122 T + p^{3} T^{2} \) |
| 59 | \( 1 + 537 T + p^{3} T^{2} \) |
| 61 | \( 1 + 950 T + p^{3} T^{2} \) |
| 67 | \( 1 + 83 T + p^{3} T^{2} \) |
| 71 | \( 1 + 180 T + p^{3} T^{2} \) |
| 73 | \( 1 - 177 T + p^{3} T^{2} \) |
| 79 | \( 1 + 220 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1112 T + p^{3} T^{2} \) |
| 89 | \( 1 - 394 T + p^{3} T^{2} \) |
| 97 | \( 1 - 826 T + p^{3} T^{2} \) |
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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
−8.868106862537499909814833785526, −8.161826612555130518738309954568, −7.22840220007416147632141059083, −6.21878468208738307763427082733, −5.77770843850026013640364384836, −4.70602494483231923046374680007, −3.35795312022273318179532010056, −1.99565030426506773879638303031, −1.66621217915994825606121072802, 0,
1.66621217915994825606121072802, 1.99565030426506773879638303031, 3.35795312022273318179532010056, 4.70602494483231923046374680007, 5.77770843850026013640364384836, 6.21878468208738307763427082733, 7.22840220007416147632141059083, 8.161826612555130518738309954568, 8.868106862537499909814833785526
