| L(s) = 1 | − 1.47·2-s + 2.48·3-s + 0.187·4-s − 3.67·6-s + 3.24·7-s + 2.68·8-s + 3.16·9-s − 4.18·11-s + 0.466·12-s + 1.78·13-s − 4.80·14-s − 4.34·16-s − 6.33·17-s − 4.68·18-s + 8.07·21-s + 6.18·22-s − 1.43·23-s + 6.65·24-s − 2.63·26-s + 0.412·27-s + 0.610·28-s + 0.339·29-s + 2.77·31-s + 1.05·32-s − 10.3·33-s + 9.36·34-s + 0.595·36-s + ⋯ |
| L(s) = 1 | − 1.04·2-s + 1.43·3-s + 0.0939·4-s − 1.49·6-s + 1.22·7-s + 0.947·8-s + 1.05·9-s − 1.26·11-s + 0.134·12-s + 0.494·13-s − 1.28·14-s − 1.08·16-s − 1.53·17-s − 1.10·18-s + 1.76·21-s + 1.31·22-s − 0.298·23-s + 1.35·24-s − 0.517·26-s + 0.0794·27-s + 0.115·28-s + 0.0630·29-s + 0.499·31-s + 0.187·32-s − 1.80·33-s + 1.60·34-s + 0.0991·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 1.47T + 2T^{2} \) |
| 3 | \( 1 - 2.48T + 3T^{2} \) |
| 7 | \( 1 - 3.24T + 7T^{2} \) |
| 11 | \( 1 + 4.18T + 11T^{2} \) |
| 13 | \( 1 - 1.78T + 13T^{2} \) |
| 17 | \( 1 + 6.33T + 17T^{2} \) |
| 23 | \( 1 + 1.43T + 23T^{2} \) |
| 29 | \( 1 - 0.339T + 29T^{2} \) |
| 31 | \( 1 - 2.77T + 31T^{2} \) |
| 37 | \( 1 + 2.70T + 37T^{2} \) |
| 41 | \( 1 - 7.13T + 41T^{2} \) |
| 43 | \( 1 + 9.89T + 43T^{2} \) |
| 47 | \( 1 - 0.445T + 47T^{2} \) |
| 53 | \( 1 + 7.23T + 53T^{2} \) |
| 59 | \( 1 + 3.14T + 59T^{2} \) |
| 61 | \( 1 + 3.06T + 61T^{2} \) |
| 67 | \( 1 - 8.55T + 67T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 73 | \( 1 - 1.82T + 73T^{2} \) |
| 79 | \( 1 + 0.698T + 79T^{2} \) |
| 83 | \( 1 - 0.552T + 83T^{2} \) |
| 89 | \( 1 + 6.82T + 89T^{2} \) |
| 97 | \( 1 + 13.2T + 97T^{2} \) |
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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
−7.72850074346895390120218832178, −7.24329462481233478008883982237, −6.22158900224471416905137638944, −5.03699461857312014520293506362, −4.59539390200084124803475248050, −3.79433130206495882163195775717, −2.72965114212717077246660316849, −2.08063494163005916180115005153, −1.41612555235352175399581837210, 0,
1.41612555235352175399581837210, 2.08063494163005916180115005153, 2.72965114212717077246660316849, 3.79433130206495882163195775717, 4.59539390200084124803475248050, 5.03699461857312014520293506362, 6.22158900224471416905137638944, 7.24329462481233478008883982237, 7.72850074346895390120218832178
