| L(s) = 1 | + 2-s + 2·3-s − 7·4-s + 2·6-s + 7·7-s − 15·8-s − 23·9-s − 8·11-s − 14·12-s − 28·13-s + 7·14-s + 41·16-s − 54·17-s − 23·18-s − 110·19-s + 14·21-s − 8·22-s − 48·23-s − 30·24-s − 28·26-s − 100·27-s − 49·28-s − 110·29-s + 12·31-s + 161·32-s − 16·33-s − 54·34-s + ⋯ |
| L(s) = 1 | + 0.353·2-s + 0.384·3-s − 7/8·4-s + 0.136·6-s + 0.377·7-s − 0.662·8-s − 0.851·9-s − 0.219·11-s − 0.336·12-s − 0.597·13-s + 0.133·14-s + 0.640·16-s − 0.770·17-s − 0.301·18-s − 1.32·19-s + 0.145·21-s − 0.0775·22-s − 0.435·23-s − 0.255·24-s − 0.211·26-s − 0.712·27-s − 0.330·28-s − 0.704·29-s + 0.0695·31-s + 0.889·32-s − 0.0844·33-s − 0.272·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 7 | \( 1 - p T \) |
| good | 2 | \( 1 - T + p^{3} T^{2} \) |
| 3 | \( 1 - 2 T + p^{3} T^{2} \) |
| 11 | \( 1 + 8 T + p^{3} T^{2} \) |
| 13 | \( 1 + 28 T + p^{3} T^{2} \) |
| 17 | \( 1 + 54 T + p^{3} T^{2} \) |
| 19 | \( 1 + 110 T + p^{3} T^{2} \) |
| 23 | \( 1 + 48 T + p^{3} T^{2} \) |
| 29 | \( 1 + 110 T + p^{3} T^{2} \) |
| 31 | \( 1 - 12 T + p^{3} T^{2} \) |
| 37 | \( 1 - 246 T + p^{3} T^{2} \) |
| 41 | \( 1 - 182 T + p^{3} T^{2} \) |
| 43 | \( 1 + 128 T + p^{3} T^{2} \) |
| 47 | \( 1 + 324 T + p^{3} T^{2} \) |
| 53 | \( 1 - 162 T + p^{3} T^{2} \) |
| 59 | \( 1 - 810 T + p^{3} T^{2} \) |
| 61 | \( 1 + 8 p T + p^{3} T^{2} \) |
| 67 | \( 1 + 244 T + p^{3} T^{2} \) |
| 71 | \( 1 + 768 T + p^{3} T^{2} \) |
| 73 | \( 1 - 702 T + p^{3} T^{2} \) |
| 79 | \( 1 - 440 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1302 T + p^{3} T^{2} \) |
| 89 | \( 1 - 730 T + p^{3} T^{2} \) |
| 97 | \( 1 + 294 T + p^{3} T^{2} \) |
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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
−11.89237034518600914822938958782, −10.81120447363476313699102904265, −9.552205077665165411705217571640, −8.678543607105515595539268343347, −7.87663652873329034579351603638, −6.22183373914071363584399207570, −5.04707921541420773464099833242, −3.96254042204938223319905892407, −2.44442207159288823608960675233, 0,
2.44442207159288823608960675233, 3.96254042204938223319905892407, 5.04707921541420773464099833242, 6.22183373914071363584399207570, 7.87663652873329034579351603638, 8.678543607105515595539268343347, 9.552205077665165411705217571640, 10.81120447363476313699102904265, 11.89237034518600914822938958782
