L(s) = 1 | − 3-s − 2·5-s + 9-s + 4·11-s + 13-s + 2·15-s − 6·17-s − 8·19-s + 8·23-s − 25-s − 27-s + 2·29-s + 8·31-s − 4·33-s − 10·37-s − 39-s + 6·41-s − 4·43-s − 2·45-s − 7·49-s + 6·51-s − 14·53-s − 8·55-s + 8·57-s − 12·59-s − 10·61-s − 2·65-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s + 1.20·11-s + 0.277·13-s + 0.516·15-s − 1.45·17-s − 1.83·19-s + 1.66·23-s − 1/5·25-s − 0.192·27-s + 0.371·29-s + 1.43·31-s − 0.696·33-s − 1.64·37-s − 0.160·39-s + 0.937·41-s − 0.609·43-s − 0.298·45-s − 49-s + 0.840·51-s − 1.92·53-s − 1.07·55-s + 1.05·57-s − 1.56·59-s − 1.28·61-s − 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−9.034750489934040733997071252152, −8.672430104572291160531976657172, −7.58509571403619500018744888863, −6.53450715872245064722726153899, −6.34363053984341960282097416121, −4.66542007417636767344834083474, −4.35038208806792458849519254326, −3.16156249407593119167913883340, −1.59378843935215793575881490850, 0,
1.59378843935215793575881490850, 3.16156249407593119167913883340, 4.35038208806792458849519254326, 4.66542007417636767344834083474, 6.34363053984341960282097416121, 6.53450715872245064722726153899, 7.58509571403619500018744888863, 8.672430104572291160531976657172, 9.034750489934040733997071252152