| L(s) = 1 | − 1.73·2-s − 3-s + 0.999·4-s − 5-s + 1.73·6-s + 1.73·8-s + 9-s + 1.73·10-s − 0.999·12-s − 3.46·13-s + 15-s − 5·16-s − 1.73·18-s + 3.46·19-s − 0.999·20-s − 1.73·24-s + 25-s + 5.99·26-s − 27-s + 3.46·29-s − 1.73·30-s − 8·31-s + 5.19·32-s + 0.999·36-s − 2·37-s − 5.99·38-s + 3.46·39-s + ⋯ |
| L(s) = 1 | − 1.22·2-s − 0.577·3-s + 0.499·4-s − 0.447·5-s + 0.707·6-s + 0.612·8-s + 0.333·9-s + 0.547·10-s − 0.288·12-s − 0.960·13-s + 0.258·15-s − 1.25·16-s − 0.408·18-s + 0.794·19-s − 0.223·20-s − 0.353·24-s + 0.200·25-s + 1.17·26-s − 0.192·27-s + 0.643·29-s − 0.316·30-s − 1.43·31-s + 0.918·32-s + 0.166·36-s − 0.328·37-s − 0.973·38-s + 0.554·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.73T + 2T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 3.46T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 3.46T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 - 6.92T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 13.8T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 3.46T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 + 17.3T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−8.967158660851930466237792642626, −8.083568253183071223840268781850, −7.38858886797075228904562480866, −6.89974843915113207165478882704, −5.65189285811065313164211046678, −4.82510184431625082527723553300, −3.92156562137037026080916460033, −2.50225461319321110788877891944, −1.17998737446998746432223357651, 0,
1.17998737446998746432223357651, 2.50225461319321110788877891944, 3.92156562137037026080916460033, 4.82510184431625082527723553300, 5.65189285811065313164211046678, 6.89974843915113207165478882704, 7.38858886797075228904562480866, 8.083568253183071223840268781850, 8.967158660851930466237792642626
