L(s) = 1 | − 2.49·2-s − 2.82·3-s + 4.22·4-s + 5-s + 7.05·6-s − 1.90·7-s − 5.55·8-s + 4.99·9-s − 2.49·10-s − 1.06·11-s − 11.9·12-s + 4.75·14-s − 2.82·15-s + 5.41·16-s + 0.637·17-s − 12.4·18-s − 5.73·19-s + 4.22·20-s + 5.38·21-s + 2.66·22-s + 3.81·23-s + 15.7·24-s + 25-s − 5.62·27-s − 8.05·28-s + 9.45·29-s + 7.05·30-s + ⋯ |
L(s) = 1 | − 1.76·2-s − 1.63·3-s + 2.11·4-s + 0.447·5-s + 2.87·6-s − 0.720·7-s − 1.96·8-s + 1.66·9-s − 0.789·10-s − 0.322·11-s − 3.44·12-s + 1.27·14-s − 0.729·15-s + 1.35·16-s + 0.154·17-s − 2.93·18-s − 1.31·19-s + 0.945·20-s + 1.17·21-s + 0.568·22-s + 0.796·23-s + 3.20·24-s + 0.200·25-s − 1.08·27-s − 1.52·28-s + 1.75·29-s + 1.28·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 2.49T + 2T^{2} \) |
| 3 | \( 1 + 2.82T + 3T^{2} \) |
| 7 | \( 1 + 1.90T + 7T^{2} \) |
| 11 | \( 1 + 1.06T + 11T^{2} \) |
| 17 | \( 1 - 0.637T + 17T^{2} \) |
| 19 | \( 1 + 5.73T + 19T^{2} \) |
| 23 | \( 1 - 3.81T + 23T^{2} \) |
| 29 | \( 1 - 9.45T + 29T^{2} \) |
| 31 | \( 1 - 1.46T + 31T^{2} \) |
| 37 | \( 1 + 0.757T + 37T^{2} \) |
| 41 | \( 1 + 0.267T + 41T^{2} \) |
| 43 | \( 1 - 0.637T + 43T^{2} \) |
| 47 | \( 1 - 9.44T + 47T^{2} \) |
| 53 | \( 1 + 6.99T + 53T^{2} \) |
| 59 | \( 1 + 0.741T + 59T^{2} \) |
| 61 | \( 1 - 4.19T + 61T^{2} \) |
| 67 | \( 1 + 8.09T + 67T^{2} \) |
| 71 | \( 1 + 9.76T + 71T^{2} \) |
| 73 | \( 1 - 3.71T + 73T^{2} \) |
| 79 | \( 1 + 9.31T + 79T^{2} \) |
| 83 | \( 1 + 5.11T + 83T^{2} \) |
| 89 | \( 1 - 12.5T + 89T^{2} \) |
| 97 | \( 1 - 4.22T + 97T^{2} \) |
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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
−9.968835118545899550969547747504, −9.105080116997518109552901283336, −8.207598342618240121616733112498, −7.02635831294781775116190016917, −6.51600577856408276116416099403, −5.82446743820835504560652361210, −4.61727994784162460116145434129, −2.68174683659477742947105495160, −1.20619947589864816191587014973, 0,
1.20619947589864816191587014973, 2.68174683659477742947105495160, 4.61727994784162460116145434129, 5.82446743820835504560652361210, 6.51600577856408276116416099403, 7.02635831294781775116190016917, 8.207598342618240121616733112498, 9.105080116997518109552901283336, 9.968835118545899550969547747504