| L(s) = 1 | − 3-s − 3·5-s − 7-s + 9-s + 2·11-s − 13-s + 3·15-s − 7·19-s + 21-s + 7·23-s + 4·25-s − 27-s + 3·29-s + 5·31-s − 2·33-s + 3·35-s + 4·37-s + 39-s − 6·41-s + 11·43-s − 3·45-s + 3·47-s + 49-s − 9·53-s − 6·55-s + 7·57-s + 8·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s − 0.377·7-s + 1/3·9-s + 0.603·11-s − 0.277·13-s + 0.774·15-s − 1.60·19-s + 0.218·21-s + 1.45·23-s + 4/5·25-s − 0.192·27-s + 0.557·29-s + 0.898·31-s − 0.348·33-s + 0.507·35-s + 0.657·37-s + 0.160·39-s − 0.937·41-s + 1.67·43-s − 0.447·45-s + 0.437·47-s + 1/7·49-s − 1.23·53-s − 0.809·55-s + 0.927·57-s + 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−7.962403144241578888444049189220, −7.19532231936169398584328524242, −6.61082539415363422253483151403, −5.95149272778342121538607358665, −4.74638414582799489889143137046, −4.35234082768866550304962513579, −3.51482721620616424464880091762, −2.56629690398633258509926568361, −1.08276969175856913160281067112, 0,
1.08276969175856913160281067112, 2.56629690398633258509926568361, 3.51482721620616424464880091762, 4.35234082768866550304962513579, 4.74638414582799489889143137046, 5.95149272778342121538607358665, 6.61082539415363422253483151403, 7.19532231936169398584328524242, 7.962403144241578888444049189220
