| L(s) = 1 | − 1.36·2-s − 0.131·4-s − 2.67·7-s + 2.91·8-s + 2.30·11-s + 3.08·13-s + 3.65·14-s − 3.71·16-s − 3.28·17-s + 5.71·19-s − 3.14·22-s − 3.43·23-s − 4.21·26-s + 0.352·28-s + 3.08·29-s − 31-s − 0.744·32-s + 4.48·34-s − 2.62·37-s − 7.81·38-s − 9.47·41-s − 8.24·43-s − 0.303·44-s + 4.69·46-s − 6.54·47-s + 0.131·49-s − 0.406·52-s + ⋯ |
| L(s) = 1 | − 0.966·2-s − 0.0659·4-s − 1.00·7-s + 1.03·8-s + 0.694·11-s + 0.855·13-s + 0.975·14-s − 0.929·16-s − 0.796·17-s + 1.31·19-s − 0.671·22-s − 0.716·23-s − 0.827·26-s + 0.0665·28-s + 0.572·29-s − 0.179·31-s − 0.131·32-s + 0.769·34-s − 0.432·37-s − 1.26·38-s − 1.47·41-s − 1.25·43-s − 0.0457·44-s + 0.692·46-s − 0.954·47-s + 0.0188·49-s − 0.0564·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 1.36T + 2T^{2} \) |
| 7 | \( 1 + 2.67T + 7T^{2} \) |
| 11 | \( 1 - 2.30T + 11T^{2} \) |
| 13 | \( 1 - 3.08T + 13T^{2} \) |
| 17 | \( 1 + 3.28T + 17T^{2} \) |
| 19 | \( 1 - 5.71T + 19T^{2} \) |
| 23 | \( 1 + 3.43T + 23T^{2} \) |
| 29 | \( 1 - 3.08T + 29T^{2} \) |
| 37 | \( 1 + 2.62T + 37T^{2} \) |
| 41 | \( 1 + 9.47T + 41T^{2} \) |
| 43 | \( 1 + 8.24T + 43T^{2} \) |
| 47 | \( 1 + 6.54T + 47T^{2} \) |
| 53 | \( 1 - 5.46T + 53T^{2} \) |
| 59 | \( 1 - 11.6T + 59T^{2} \) |
| 61 | \( 1 + 6.98T + 61T^{2} \) |
| 67 | \( 1 - 13.6T + 67T^{2} \) |
| 71 | \( 1 - 8.78T + 71T^{2} \) |
| 73 | \( 1 + 7.19T + 73T^{2} \) |
| 79 | \( 1 + 16.9T + 79T^{2} \) |
| 83 | \( 1 + 0.438T + 83T^{2} \) |
| 89 | \( 1 - 11.6T + 89T^{2} \) |
| 97 | \( 1 - 0.582T + 97T^{2} \) |
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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
−7.75378629041869105275108577012, −6.80912530916373965794614913912, −6.58068119800574687032826265191, −5.55107147450377071373034274856, −4.73633360655546950351959524753, −3.78901201338878805673251618663, −3.28232836187498748487507075963, −1.97314057615125465204805071600, −1.08073215586009723752697423793, 0,
1.08073215586009723752697423793, 1.97314057615125465204805071600, 3.28232836187498748487507075963, 3.78901201338878805673251618663, 4.73633360655546950351959524753, 5.55107147450377071373034274856, 6.58068119800574687032826265191, 6.80912530916373965794614913912, 7.75378629041869105275108577012
