| L(s) = 1 | − 1.73·2-s − 2.73·3-s + 0.999·4-s + 1.73·5-s + 4.73·6-s + 4.73·7-s + 1.73·8-s + 4.46·9-s − 2.99·10-s + 11-s − 2.73·12-s − 8.19·14-s − 4.73·15-s − 5·16-s − 4.26·17-s − 7.73·18-s − 1.26·19-s + 1.73·20-s − 12.9·21-s − 1.73·22-s − 3.46·23-s − 4.73·24-s − 2.00·25-s − 3.99·27-s + 4.73·28-s + 4.26·29-s + 8.19·30-s + ⋯ |
| L(s) = 1 | − 1.22·2-s − 1.57·3-s + 0.499·4-s + 0.774·5-s + 1.93·6-s + 1.78·7-s + 0.612·8-s + 1.48·9-s − 0.948·10-s + 0.301·11-s − 0.788·12-s − 2.19·14-s − 1.22·15-s − 1.25·16-s − 1.03·17-s − 1.82·18-s − 0.290·19-s + 0.387·20-s − 2.82·21-s − 0.369·22-s − 0.722·23-s − 0.965·24-s − 0.400·25-s − 0.769·27-s + 0.894·28-s + 0.792·29-s + 1.49·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.73T + 2T^{2} \) |
| 3 | \( 1 + 2.73T + 3T^{2} \) |
| 5 | \( 1 - 1.73T + 5T^{2} \) |
| 7 | \( 1 - 4.73T + 7T^{2} \) |
| 17 | \( 1 + 4.26T + 17T^{2} \) |
| 19 | \( 1 + 1.26T + 19T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 - 4.26T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 + 1.73T + 37T^{2} \) |
| 41 | \( 1 + 9T + 41T^{2} \) |
| 43 | \( 1 + 12.3T + 43T^{2} \) |
| 47 | \( 1 - 4.73T + 47T^{2} \) |
| 53 | \( 1 + 0.464T + 53T^{2} \) |
| 59 | \( 1 + 1.26T + 59T^{2} \) |
| 61 | \( 1 + 7.19T + 61T^{2} \) |
| 67 | \( 1 + 4.73T + 67T^{2} \) |
| 71 | \( 1 + 3.46T + 71T^{2} \) |
| 73 | \( 1 + 12.4T + 73T^{2} \) |
| 79 | \( 1 + 4.19T + 79T^{2} \) |
| 83 | \( 1 - 9.12T + 83T^{2} \) |
| 89 | \( 1 + 3.46T + 89T^{2} \) |
| 97 | \( 1 - 3.46T + 97T^{2} \) |
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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
−8.804775764886987705445094143874, −8.233816182281546400996346950363, −7.28209130680040409253173349116, −6.54934328305350283824644262972, −5.64233885191080285053443132873, −4.88831870263982615052864886581, −4.30703374536162876605522674923, −1.92861885875819579693784452552, −1.44958012664702838224256496010, 0,
1.44958012664702838224256496010, 1.92861885875819579693784452552, 4.30703374536162876605522674923, 4.88831870263982615052864886581, 5.64233885191080285053443132873, 6.54934328305350283824644262972, 7.28209130680040409253173349116, 8.233816182281546400996346950363, 8.804775764886987705445094143874
