| L(s) = 1 | − 1.73·2-s + 0.999·4-s − 7-s + 1.73·8-s + 3.46·11-s − 2·13-s + 1.73·14-s − 5·16-s − 3.46·17-s − 4·19-s − 5.99·22-s + 3.46·23-s + 3.46·26-s − 0.999·28-s − 4·31-s + 5.19·32-s + 5.99·34-s − 2·37-s + 6.92·38-s + 10.3·41-s + 4·43-s + 3.46·44-s − 5.99·46-s − 6.92·47-s + 49-s − 1.99·52-s + 6.92·53-s + ⋯ |
| L(s) = 1 | − 1.22·2-s + 0.499·4-s − 0.377·7-s + 0.612·8-s + 1.04·11-s − 0.554·13-s + 0.462·14-s − 1.25·16-s − 0.840·17-s − 0.917·19-s − 1.27·22-s + 0.722·23-s + 0.679·26-s − 0.188·28-s − 0.718·31-s + 0.918·32-s + 1.02·34-s − 0.328·37-s + 1.12·38-s + 1.62·41-s + 0.609·43-s + 0.522·44-s − 0.884·46-s − 1.01·47-s + 0.142·49-s − 0.277·52-s + 0.951·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 2 | \( 1 + 1.73T + 2T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 3.46T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 - 6.92T + 53T^{2} \) |
| 59 | \( 1 + 6.92T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 + 14T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 3.46T + 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−9.152802080925025956854116387692, −8.486047191146135179383478045838, −7.48318071343520741736810069807, −6.87853029702903180920342868933, −6.05420919147302561108896002088, −4.72453124273459494740236995495, −3.97601938448959508043292558696, −2.55220020995291457056915011569, −1.41141467297432757952072339356, 0,
1.41141467297432757952072339356, 2.55220020995291457056915011569, 3.97601938448959508043292558696, 4.72453124273459494740236995495, 6.05420919147302561108896002088, 6.87853029702903180920342868933, 7.48318071343520741736810069807, 8.486047191146135179383478045838, 9.152802080925025956854116387692
