| L(s) = 1 | − 3·3-s − 5-s − 7-s + 6·9-s + 3·13-s + 3·15-s + 17-s + 6·19-s + 3·21-s − 6·23-s + 25-s − 9·27-s + 9·29-s + 4·31-s + 35-s + 2·37-s − 9·39-s + 4·41-s + 10·43-s − 6·45-s + 47-s + 49-s − 3·51-s + 4·53-s − 18·57-s + 8·59-s + 8·61-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.447·5-s − 0.377·7-s + 2·9-s + 0.832·13-s + 0.774·15-s + 0.242·17-s + 1.37·19-s + 0.654·21-s − 1.25·23-s + 1/5·25-s − 1.73·27-s + 1.67·29-s + 0.718·31-s + 0.169·35-s + 0.328·37-s − 1.44·39-s + 0.624·41-s + 1.52·43-s − 0.894·45-s + 0.145·47-s + 1/7·49-s − 0.420·51-s + 0.549·53-s − 2.38·57-s + 1.04·59-s + 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−14.30435539861844, −13.85679315030498, −13.35466169232878, −12.73644397135482, −12.16171910258482, −11.93797710050588, −11.54432510642474, −10.93471795148507, −10.48085332252764, −10.03561544931463, −9.567601051223175, −8.832891820594666, −8.139317539481562, −7.630251028153458, −7.037878653143269, −6.494753718450833, −6.022144720873075, −5.597852982604246, −5.055476756225529, −4.253912622375719, −4.034885886463491, −3.139779460092931, −2.411505266681509, −1.110323420398137, −0.9923063619578272, 0,
0.9923063619578272, 1.110323420398137, 2.411505266681509, 3.139779460092931, 4.034885886463491, 4.253912622375719, 5.055476756225529, 5.597852982604246, 6.022144720873075, 6.494753718450833, 7.037878653143269, 7.630251028153458, 8.139317539481562, 8.832891820594666, 9.567601051223175, 10.03561544931463, 10.48085332252764, 10.93471795148507, 11.54432510642474, 11.93797710050588, 12.16171910258482, 12.73644397135482, 13.35466169232878, 13.85679315030498, 14.30435539861844
