| L(s) = 1 | + 2-s + 4-s − 4·5-s + 7-s + 8-s − 4·10-s + 2·11-s + 14-s + 16-s − 8·17-s + 19-s − 4·20-s + 2·22-s + 6·23-s + 11·25-s + 28-s + 2·29-s − 8·31-s + 32-s − 8·34-s − 4·35-s − 10·37-s + 38-s − 4·40-s − 2·41-s − 8·43-s + 2·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.78·5-s + 0.377·7-s + 0.353·8-s − 1.26·10-s + 0.603·11-s + 0.267·14-s + 1/4·16-s − 1.94·17-s + 0.229·19-s − 0.894·20-s + 0.426·22-s + 1.25·23-s + 11/5·25-s + 0.188·28-s + 0.371·29-s − 1.43·31-s + 0.176·32-s − 1.37·34-s − 0.676·35-s − 1.64·37-s + 0.162·38-s − 0.632·40-s − 0.312·41-s − 1.21·43-s + 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−8.737832900229588979192500298009, −7.58407312108859362556917900199, −7.07042870092215796696132683582, −6.42687391708333947396758328239, −5.02997148868531024713175067869, −4.57867054664356439103294666774, −3.74519823693831207291229286186, −3.07520607901890531511748990154, −1.65233750177168927141774047496, 0,
1.65233750177168927141774047496, 3.07520607901890531511748990154, 3.74519823693831207291229286186, 4.57867054664356439103294666774, 5.02997148868531024713175067869, 6.42687391708333947396758328239, 7.07042870092215796696132683582, 7.58407312108859362556917900199, 8.737832900229588979192500298009
