L(s) = 1 | + 2·2-s + 2·3-s + 2·4-s − 2·5-s + 4·6-s + 9-s − 4·10-s − 4·11-s + 4·12-s − 2·13-s − 4·15-s − 4·16-s − 3·17-s + 2·18-s − 7·19-s − 4·20-s − 8·22-s + 9·23-s − 25-s − 4·26-s − 4·27-s − 5·29-s − 8·30-s + 10·31-s − 8·32-s − 8·33-s − 6·34-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 1.15·3-s + 4-s − 0.894·5-s + 1.63·6-s + 1/3·9-s − 1.26·10-s − 1.20·11-s + 1.15·12-s − 0.554·13-s − 1.03·15-s − 16-s − 0.727·17-s + 0.471·18-s − 1.60·19-s − 0.894·20-s − 1.70·22-s + 1.87·23-s − 1/5·25-s − 0.784·26-s − 0.769·27-s − 0.928·29-s − 1.46·30-s + 1.79·31-s − 1.41·32-s − 1.39·33-s − 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3283 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3283 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 67 | \( 1 - T \) |
good | 2 | \( 1 - p T + p T^{2} \) |
| 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−8.320726214151838262283420379343, −7.44736975796997969473564803898, −6.81376357941682546326237872044, −5.81475989788622002802976993893, −4.87570582366702931530848039294, −4.35262807617858586010932017113, −3.54244384420252293675307773307, −2.75494524842014520334026530020, −2.27649351271661095450212733064, 0,
2.27649351271661095450212733064, 2.75494524842014520334026530020, 3.54244384420252293675307773307, 4.35262807617858586010932017113, 4.87570582366702931530848039294, 5.81475989788622002802976993893, 6.81376357941682546326237872044, 7.44736975796997969473564803898, 8.320726214151838262283420379343