| L(s) = 1 | + 3.80·2-s + 3·3-s + 6.49·4-s + 3.12·5-s + 11.4·6-s − 5.73·8-s + 9·9-s + 11.8·10-s − 22.0·11-s + 19.4·12-s + 13·13-s + 9.36·15-s − 73.7·16-s + 9.29·17-s + 34.2·18-s − 120.·19-s + 20.2·20-s − 84.0·22-s + 142.·23-s − 17.2·24-s − 115.·25-s + 49.4·26-s + 27·27-s − 299.·29-s + 35.6·30-s − 52.7·31-s − 234.·32-s + ⋯ |
| L(s) = 1 | + 1.34·2-s + 0.577·3-s + 0.811·4-s + 0.279·5-s + 0.777·6-s − 0.253·8-s + 0.333·9-s + 0.375·10-s − 0.605·11-s + 0.468·12-s + 0.277·13-s + 0.161·15-s − 1.15·16-s + 0.132·17-s + 0.448·18-s − 1.45·19-s + 0.226·20-s − 0.814·22-s + 1.28·23-s − 0.146·24-s − 0.922·25-s + 0.373·26-s + 0.192·27-s − 1.91·29-s + 0.216·30-s − 0.305·31-s − 1.29·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - 13T \) |
| good | 2 | \( 1 - 3.80T + 8T^{2} \) |
| 5 | \( 1 - 3.12T + 125T^{2} \) |
| 11 | \( 1 + 22.0T + 1.33e3T^{2} \) |
| 17 | \( 1 - 9.29T + 4.91e3T^{2} \) |
| 19 | \( 1 + 120.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 142.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 299.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 52.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 169.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 40.7T + 6.89e4T^{2} \) |
| 43 | \( 1 - 240.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 398.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 315.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 58.0T + 2.05e5T^{2} \) |
| 61 | \( 1 - 563.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 329.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 610.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 816.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.02e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.42e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 595.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 374.T + 9.12e5T^{2} \) |
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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
−8.534542578783395019275415507149, −7.49907961817281965354793652945, −6.71874562657858517778967904872, −5.80471736851999485641363743673, −5.20434170391312044836034609054, −4.22390772819132650917237013329, −3.57179670366677317467078267616, −2.63472472066021368431823978482, −1.80235813760227871159486424818, 0,
1.80235813760227871159486424818, 2.63472472066021368431823978482, 3.57179670366677317467078267616, 4.22390772819132650917237013329, 5.20434170391312044836034609054, 5.80471736851999485641363743673, 6.71874562657858517778967904872, 7.49907961817281965354793652945, 8.534542578783395019275415507149
