| L(s) = 1 | + 2-s − 7·4-s + 9·7-s − 15·8-s − 11·11-s − 2·13-s + 9·14-s + 41·16-s + 21·17-s − 85·19-s − 11·22-s + 22·23-s − 2·26-s − 63·28-s + 165·29-s − 83·31-s + 161·32-s + 21·34-s − 37-s − 85·38-s + 478·41-s + 8·43-s + 77·44-s + 22·46-s + 126·47-s − 262·49-s + 14·52-s + ⋯ |
| L(s) = 1 | + 0.353·2-s − 7/8·4-s + 0.485·7-s − 0.662·8-s − 0.301·11-s − 0.0426·13-s + 0.171·14-s + 0.640·16-s + 0.299·17-s − 1.02·19-s − 0.106·22-s + 0.199·23-s − 0.0150·26-s − 0.425·28-s + 1.05·29-s − 0.480·31-s + 0.889·32-s + 0.105·34-s − 0.00444·37-s − 0.362·38-s + 1.82·41-s + 0.0283·43-s + 0.263·44-s + 0.0705·46-s + 0.391·47-s − 0.763·49-s + 0.0373·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + p T \) |
| good | 2 | \( 1 - T + p^{3} T^{2} \) |
| 7 | \( 1 - 9 T + p^{3} T^{2} \) |
| 13 | \( 1 + 2 T + p^{3} T^{2} \) |
| 17 | \( 1 - 21 T + p^{3} T^{2} \) |
| 19 | \( 1 + 85 T + p^{3} T^{2} \) |
| 23 | \( 1 - 22 T + p^{3} T^{2} \) |
| 29 | \( 1 - 165 T + p^{3} T^{2} \) |
| 31 | \( 1 + 83 T + p^{3} T^{2} \) |
| 37 | \( 1 + T + p^{3} T^{2} \) |
| 41 | \( 1 - 478 T + p^{3} T^{2} \) |
| 43 | \( 1 - 8 T + p^{3} T^{2} \) |
| 47 | \( 1 - 126 T + p^{3} T^{2} \) |
| 53 | \( 1 + 683 T + p^{3} T^{2} \) |
| 59 | \( 1 - 290 T + p^{3} T^{2} \) |
| 61 | \( 1 - 257 T + p^{3} T^{2} \) |
| 67 | \( 1 + 776 T + p^{3} T^{2} \) |
| 71 | \( 1 - 313 T + p^{3} T^{2} \) |
| 73 | \( 1 + 902 T + p^{3} T^{2} \) |
| 79 | \( 1 - 830 T + p^{3} T^{2} \) |
| 83 | \( 1 - 842 T + p^{3} T^{2} \) |
| 89 | \( 1 + 25 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1784 T + p^{3} T^{2} \) |
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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
−8.210093440370200192543029176828, −7.62256304063891201212869279331, −6.47841277878905766090412499460, −5.75569391616672981862649410333, −4.87648956170411979216116399187, −4.35972968137231585033528718563, −3.40955250903035554453836603867, −2.40842781796527223016994111182, −1.12358224923234046564521598408, 0,
1.12358224923234046564521598408, 2.40842781796527223016994111182, 3.40955250903035554453836603867, 4.35972968137231585033528718563, 4.87648956170411979216116399187, 5.75569391616672981862649410333, 6.47841277878905766090412499460, 7.62256304063891201212869279331, 8.210093440370200192543029176828
