| L(s) = 1 | − 2.66·2-s + 7.39·3-s − 0.909·4-s + 10.9·5-s − 19.7·6-s − 23.1·7-s + 23.7·8-s + 27.7·9-s − 29.0·10-s + 11·11-s − 6.73·12-s + 61.6·14-s + 80.8·15-s − 55.8·16-s − 14.1·17-s − 73.8·18-s + 41.3·19-s − 9.94·20-s − 171.·21-s − 29.2·22-s − 55.5·23-s + 175.·24-s − 5.65·25-s + 5.47·27-s + 21.0·28-s + 114.·29-s − 215.·30-s + ⋯ |
| L(s) = 1 | − 0.941·2-s + 1.42·3-s − 0.113·4-s + 0.977·5-s − 1.34·6-s − 1.25·7-s + 1.04·8-s + 1.02·9-s − 0.919·10-s + 0.301·11-s − 0.161·12-s + 1.17·14-s + 1.39·15-s − 0.873·16-s − 0.201·17-s − 0.967·18-s + 0.499·19-s − 0.111·20-s − 1.78·21-s − 0.283·22-s − 0.503·23-s + 1.49·24-s − 0.0452·25-s + 0.0390·27-s + 0.142·28-s + 0.733·29-s − 1.30·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 - 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.66T + 8T^{2} \) |
| 3 | \( 1 - 7.39T + 27T^{2} \) |
| 5 | \( 1 - 10.9T + 125T^{2} \) |
| 7 | \( 1 + 23.1T + 343T^{2} \) |
| 17 | \( 1 + 14.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 41.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 55.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 114.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 84.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + 144.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 431.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 548.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 493.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 116.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 493.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 280.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.01e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 264.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 525.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 747.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 520.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 982.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 613.T + 9.12e5T^{2} \) |
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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
−8.594859412558324062622341150524, −8.072056692481916396156062036513, −7.09111918073717752853906324964, −6.39194907535684924313915612297, −5.28585534921406298482836566337, −4.03651249891544089472551377802, −3.22620354048132594189303676136, −2.27819547161983890164519835217, −1.40225172610172075632885093613, 0,
1.40225172610172075632885093613, 2.27819547161983890164519835217, 3.22620354048132594189303676136, 4.03651249891544089472551377802, 5.28585534921406298482836566337, 6.39194907535684924313915612297, 7.09111918073717752853906324964, 8.072056692481916396156062036513, 8.594859412558324062622341150524
