| L(s) = 1 | − 5·2-s + 17·4-s + 6·5-s + 7·7-s − 45·8-s − 30·10-s + 11·11-s + 70·13-s − 35·14-s + 89·16-s − 126·17-s − 80·19-s + 102·20-s − 55·22-s + 200·23-s − 89·25-s − 350·26-s + 119·28-s − 134·29-s − 244·31-s − 85·32-s + 630·34-s + 42·35-s − 314·37-s + 400·38-s − 270·40-s − 278·41-s + ⋯ |
| L(s) = 1 | − 1.76·2-s + 17/8·4-s + 0.536·5-s + 0.377·7-s − 1.98·8-s − 0.948·10-s + 0.301·11-s + 1.49·13-s − 0.668·14-s + 1.39·16-s − 1.79·17-s − 0.965·19-s + 1.14·20-s − 0.533·22-s + 1.81·23-s − 0.711·25-s − 2.64·26-s + 0.803·28-s − 0.858·29-s − 1.41·31-s − 0.469·32-s + 3.17·34-s + 0.202·35-s − 1.39·37-s + 1.70·38-s − 1.06·40-s − 1.05·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - p T \) |
| 11 | \( 1 - p T \) |
| good | 2 | \( 1 + 5 T + p^{3} T^{2} \) |
| 5 | \( 1 - 6 T + p^{3} T^{2} \) |
| 13 | \( 1 - 70 T + p^{3} T^{2} \) |
| 17 | \( 1 + 126 T + p^{3} T^{2} \) |
| 19 | \( 1 + 80 T + p^{3} T^{2} \) |
| 23 | \( 1 - 200 T + p^{3} T^{2} \) |
| 29 | \( 1 + 134 T + p^{3} T^{2} \) |
| 31 | \( 1 + 244 T + p^{3} T^{2} \) |
| 37 | \( 1 + 314 T + p^{3} T^{2} \) |
| 41 | \( 1 + 278 T + p^{3} T^{2} \) |
| 43 | \( 1 + 372 T + p^{3} T^{2} \) |
| 47 | \( 1 - 84 T + p^{3} T^{2} \) |
| 53 | \( 1 + 182 T + p^{3} T^{2} \) |
| 59 | \( 1 - 756 T + p^{3} T^{2} \) |
| 61 | \( 1 - 694 T + p^{3} T^{2} \) |
| 67 | \( 1 - 820 T + p^{3} T^{2} \) |
| 71 | \( 1 + 160 T + p^{3} T^{2} \) |
| 73 | \( 1 + 2 T + p^{3} T^{2} \) |
| 79 | \( 1 - 40 T + p^{3} T^{2} \) |
| 83 | \( 1 + 760 T + p^{3} T^{2} \) |
| 89 | \( 1 - 102 T + p^{3} T^{2} \) |
| 97 | \( 1 + 862 T + p^{3} T^{2} \) |
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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
−9.408712110222178681644966832017, −8.726448995496116423768774621286, −8.394101912253338922729784726265, −6.96200017848911899796799825238, −6.61522470981982287389387365240, −5.36436104186987348344681515303, −3.76925457717928999327392914948, −2.16794736388765132657509882704, −1.44895167244467613910571674105, 0,
1.44895167244467613910571674105, 2.16794736388765132657509882704, 3.76925457717928999327392914948, 5.36436104186987348344681515303, 6.61522470981982287389387365240, 6.96200017848911899796799825238, 8.394101912253338922729784726265, 8.726448995496116423768774621286, 9.408712110222178681644966832017
