| L(s) = 1 | + 2-s + 2.26·3-s + 4-s + 3.67·5-s + 2.26·6-s + 8-s + 2.12·9-s + 3.67·10-s − 2.32·11-s + 2.26·12-s + 5.27·13-s + 8.32·15-s + 16-s − 4.70·17-s + 2.12·18-s + 5.94·19-s + 3.67·20-s − 2.32·22-s − 8.65·23-s + 2.26·24-s + 8.52·25-s + 5.27·26-s − 1.97·27-s − 29-s + 8.32·30-s − 1.97·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.30·3-s + 0.5·4-s + 1.64·5-s + 0.924·6-s + 0.353·8-s + 0.708·9-s + 1.16·10-s − 0.701·11-s + 0.653·12-s + 1.46·13-s + 2.15·15-s + 0.250·16-s − 1.14·17-s + 0.501·18-s + 1.36·19-s + 0.822·20-s − 0.496·22-s − 1.80·23-s + 0.462·24-s + 1.70·25-s + 1.03·26-s − 0.380·27-s − 0.185·29-s + 1.52·30-s − 0.355·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.036530030\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.036530030\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - 2.26T + 3T^{2} \) |
| 5 | \( 1 - 3.67T + 5T^{2} \) |
| 11 | \( 1 + 2.32T + 11T^{2} \) |
| 13 | \( 1 - 5.27T + 13T^{2} \) |
| 17 | \( 1 + 4.70T + 17T^{2} \) |
| 19 | \( 1 - 5.94T + 19T^{2} \) |
| 23 | \( 1 + 8.65T + 23T^{2} \) |
| 31 | \( 1 + 1.97T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 + 12.1T + 43T^{2} \) |
| 47 | \( 1 + 1.97T + 47T^{2} \) |
| 53 | \( 1 - 6.32T + 53T^{2} \) |
| 59 | \( 1 + 7.17T + 59T^{2} \) |
| 61 | \( 1 + 11.1T + 61T^{2} \) |
| 67 | \( 1 + 2.25T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 7.17T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 - 5.83T + 89T^{2} \) |
| 97 | \( 1 - 4.13T + 97T^{2} \) |
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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
−8.824098158602391462147150508726, −8.111383487565450543980107704022, −7.29760906536792464566107571317, −6.19180138400295040733021188430, −5.86773572402533708297973346467, −4.91287888083819033011222868080, −3.82118944347297926172892815682, −3.05172922911539021457222695358, −2.21297219260192775001974318205, −1.60516856580702601632244991063,
1.60516856580702601632244991063, 2.21297219260192775001974318205, 3.05172922911539021457222695358, 3.82118944347297926172892815682, 4.91287888083819033011222868080, 5.86773572402533708297973346467, 6.19180138400295040733021188430, 7.29760906536792464566107571317, 8.111383487565450543980107704022, 8.824098158602391462147150508726
