| L(s) = 1 | − 2-s + 4-s − 5-s − 2·7-s − 8-s + 10-s + 4·13-s + 2·14-s + 16-s + 17-s − 8·19-s − 20-s + 25-s − 4·26-s − 2·28-s + 6·29-s + 8·31-s − 32-s − 34-s + 2·35-s + 2·37-s + 8·38-s + 40-s + 10·43-s − 3·49-s − 50-s + 4·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.755·7-s − 0.353·8-s + 0.316·10-s + 1.10·13-s + 0.534·14-s + 1/4·16-s + 0.242·17-s − 1.83·19-s − 0.223·20-s + 1/5·25-s − 0.784·26-s − 0.377·28-s + 1.11·29-s + 1.43·31-s − 0.176·32-s − 0.171·34-s + 0.338·35-s + 0.328·37-s + 1.29·38-s + 0.158·40-s + 1.52·43-s − 3/7·49-s − 0.141·50-s + 0.554·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 185130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.545619144\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.545619144\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−13.11920330639014, −12.54022280781524, −12.27577081405146, −11.57547017092642, −11.23885640917686, −10.69869753563484, −10.18642632836833, −10.03533843987558, −9.201978280755661, −8.809114347143868, −8.394401428932854, −8.049981178468351, −7.419180708981762, −6.770833797943427, −6.419662461174497, −6.075855587069732, −5.476707814366508, −4.521125713508704, −4.231900405331392, −3.607507954430606, −2.947357404216901, −2.520800219485085, −1.771728203456030, −0.9324293804308260, −0.4963769022976066,
0.4963769022976066, 0.9324293804308260, 1.771728203456030, 2.520800219485085, 2.947357404216901, 3.607507954430606, 4.231900405331392, 4.521125713508704, 5.476707814366508, 6.075855587069732, 6.419662461174497, 6.770833797943427, 7.419180708981762, 8.049981178468351, 8.394401428932854, 8.809114347143868, 9.201978280755661, 10.03533843987558, 10.18642632836833, 10.69869753563484, 11.23885640917686, 11.57547017092642, 12.27577081405146, 12.54022280781524, 13.11920330639014
