| L(s) = 1 | − 1.05·2-s − 0.881·4-s + 1.01·7-s + 3.04·8-s − 5.12·11-s + 6.08·13-s − 1.07·14-s − 1.45·16-s − 3.19·17-s + 3.42·19-s + 5.41·22-s − 2.91·23-s − 6.43·26-s − 0.892·28-s + 1.55·29-s − 7.99·31-s − 4.55·32-s + 3.37·34-s + 8.40·37-s − 3.62·38-s + 1.86·41-s − 5.22·43-s + 4.51·44-s + 3.08·46-s + 4.80·47-s − 5.97·49-s − 5.36·52-s + ⋯ |
| L(s) = 1 | − 0.747·2-s − 0.440·4-s + 0.382·7-s + 1.07·8-s − 1.54·11-s + 1.68·13-s − 0.285·14-s − 0.364·16-s − 0.774·17-s + 0.786·19-s + 1.15·22-s − 0.608·23-s − 1.26·26-s − 0.168·28-s + 0.288·29-s − 1.43·31-s − 0.804·32-s + 0.579·34-s + 1.38·37-s − 0.588·38-s + 0.291·41-s − 0.796·43-s + 0.680·44-s + 0.455·46-s + 0.700·47-s − 0.853·49-s − 0.744·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 2 | \( 1 + 1.05T + 2T^{2} \) |
| 7 | \( 1 - 1.01T + 7T^{2} \) |
| 11 | \( 1 + 5.12T + 11T^{2} \) |
| 13 | \( 1 - 6.08T + 13T^{2} \) |
| 17 | \( 1 + 3.19T + 17T^{2} \) |
| 19 | \( 1 - 3.42T + 19T^{2} \) |
| 23 | \( 1 + 2.91T + 23T^{2} \) |
| 29 | \( 1 - 1.55T + 29T^{2} \) |
| 31 | \( 1 + 7.99T + 31T^{2} \) |
| 37 | \( 1 - 8.40T + 37T^{2} \) |
| 41 | \( 1 - 1.86T + 41T^{2} \) |
| 43 | \( 1 + 5.22T + 43T^{2} \) |
| 47 | \( 1 - 4.80T + 47T^{2} \) |
| 53 | \( 1 + 10.0T + 53T^{2} \) |
| 59 | \( 1 + 2.89T + 59T^{2} \) |
| 61 | \( 1 + 2.30T + 61T^{2} \) |
| 67 | \( 1 - 4.64T + 67T^{2} \) |
| 71 | \( 1 - 7.73T + 71T^{2} \) |
| 73 | \( 1 + 0.595T + 73T^{2} \) |
| 79 | \( 1 - 11.0T + 79T^{2} \) |
| 83 | \( 1 + 14.2T + 83T^{2} \) |
| 89 | \( 1 - 6.39T + 89T^{2} \) |
| 97 | \( 1 + 14.0T + 97T^{2} \) |
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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
−8.006775202260345255584725900443, −7.40764284391993895487882364743, −6.37721753828698178572378517686, −5.57326407403375673112657938699, −4.91216989324187205803041668338, −4.11720617280295027456430976817, −3.26032550262072674384587819086, −2.10916183614382232714632810257, −1.17042211778982174350378315240, 0,
1.17042211778982174350378315240, 2.10916183614382232714632810257, 3.26032550262072674384587819086, 4.11720617280295027456430976817, 4.91216989324187205803041668338, 5.57326407403375673112657938699, 6.37721753828698178572378517686, 7.40764284391993895487882364743, 8.006775202260345255584725900443
