| L(s) = 1 | + 2.68·2-s + 3.12·3-s + 5.20·4-s − 2.33·5-s + 8.39·6-s − 0.266·7-s + 8.58·8-s + 6.77·9-s − 6.26·10-s − 11-s + 16.2·12-s − 0.715·14-s − 7.30·15-s + 12.6·16-s − 1.53·17-s + 18.1·18-s − 7.55·19-s − 12.1·20-s − 0.833·21-s − 2.68·22-s − 3.95·23-s + 26.8·24-s + 0.451·25-s + 11.8·27-s − 1.38·28-s + 1.08·29-s − 19.5·30-s + ⋯ |
| L(s) = 1 | + 1.89·2-s + 1.80·3-s + 2.60·4-s − 1.04·5-s + 3.42·6-s − 0.100·7-s + 3.03·8-s + 2.25·9-s − 1.98·10-s − 0.301·11-s + 4.69·12-s − 0.191·14-s − 1.88·15-s + 3.16·16-s − 0.372·17-s + 4.28·18-s − 1.73·19-s − 2.71·20-s − 0.181·21-s − 0.572·22-s − 0.824·23-s + 5.48·24-s + 0.0902·25-s + 2.27·27-s − 0.262·28-s + 0.201·29-s − 3.57·30-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
| \(L(1)\) |
\(\approx\) |
\(8.540072057\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.540072057\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.68T + 2T^{2} \) |
| 3 | \( 1 - 3.12T + 3T^{2} \) |
| 5 | \( 1 + 2.33T + 5T^{2} \) |
| 7 | \( 1 + 0.266T + 7T^{2} \) |
| 17 | \( 1 + 1.53T + 17T^{2} \) |
| 19 | \( 1 + 7.55T + 19T^{2} \) |
| 23 | \( 1 + 3.95T + 23T^{2} \) |
| 29 | \( 1 - 1.08T + 29T^{2} \) |
| 31 | \( 1 - 3.25T + 31T^{2} \) |
| 37 | \( 1 + 3.44T + 37T^{2} \) |
| 41 | \( 1 - 6.30T + 41T^{2} \) |
| 43 | \( 1 - 2.45T + 43T^{2} \) |
| 47 | \( 1 - 6.48T + 47T^{2} \) |
| 53 | \( 1 - 5.06T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 - 9.66T + 61T^{2} \) |
| 67 | \( 1 + 6.41T + 67T^{2} \) |
| 71 | \( 1 + 13.0T + 71T^{2} \) |
| 73 | \( 1 - 1.49T + 73T^{2} \) |
| 79 | \( 1 + 3.63T + 79T^{2} \) |
| 83 | \( 1 - 9.17T + 83T^{2} \) |
| 89 | \( 1 - 8.10T + 89T^{2} \) |
| 97 | \( 1 - 12.7T + 97T^{2} \) |
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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
−8.986964456400132549057964250555, −8.145659376561199785481940746386, −7.63744207260411683279650210393, −6.86758564294689255170227488060, −6.01611786512826575887102822277, −4.57623016367105376307765974713, −4.18154221929242423571824485298, −3.51247173795628311509547063787, −2.65997262697985731238392580547, −1.96229998571087458057127446777,
1.96229998571087458057127446777, 2.65997262697985731238392580547, 3.51247173795628311509547063787, 4.18154221929242423571824485298, 4.57623016367105376307765974713, 6.01611786512826575887102822277, 6.86758564294689255170227488060, 7.63744207260411683279650210393, 8.145659376561199785481940746386, 8.986964456400132549057964250555
