| L(s) = 1 | − 2.24·2-s + 3.04·4-s − 3.27·5-s − 2.35·8-s + 7.35·10-s − 0.445·11-s + 4.44·13-s − 0.801·16-s + 0.648·17-s − 1.16·19-s − 9.97·20-s + 22-s − 4.35·23-s + 5.71·25-s − 9.97·26-s + 2.13·29-s − 6.18·31-s + 6.51·32-s − 1.45·34-s − 7.65·37-s + 2.62·38-s + 7.71·40-s + 3.56·41-s − 6.02·43-s − 1.35·44-s + 9.78·46-s + 12.7·47-s + ⋯ |
| L(s) = 1 | − 1.58·2-s + 1.52·4-s − 1.46·5-s − 0.833·8-s + 2.32·10-s − 0.134·11-s + 1.23·13-s − 0.200·16-s + 0.157·17-s − 0.268·19-s − 2.23·20-s + 0.213·22-s − 0.908·23-s + 1.14·25-s − 1.95·26-s + 0.396·29-s − 1.11·31-s + 1.15·32-s − 0.249·34-s − 1.25·37-s + 0.425·38-s + 1.21·40-s + 0.556·41-s − 0.918·43-s − 0.204·44-s + 1.44·46-s + 1.85·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3087 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3087 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 2.24T + 2T^{2} \) |
| 5 | \( 1 + 3.27T + 5T^{2} \) |
| 11 | \( 1 + 0.445T + 11T^{2} \) |
| 13 | \( 1 - 4.44T + 13T^{2} \) |
| 17 | \( 1 - 0.648T + 17T^{2} \) |
| 19 | \( 1 + 1.16T + 19T^{2} \) |
| 23 | \( 1 + 4.35T + 23T^{2} \) |
| 29 | \( 1 - 2.13T + 29T^{2} \) |
| 31 | \( 1 + 6.18T + 31T^{2} \) |
| 37 | \( 1 + 7.65T + 37T^{2} \) |
| 41 | \( 1 - 3.56T + 41T^{2} \) |
| 43 | \( 1 + 6.02T + 43T^{2} \) |
| 47 | \( 1 - 12.7T + 47T^{2} \) |
| 53 | \( 1 + 0.594T + 53T^{2} \) |
| 59 | \( 1 - 7.06T + 59T^{2} \) |
| 61 | \( 1 - 13.2T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 + 4.29T + 71T^{2} \) |
| 73 | \( 1 + 5.08T + 73T^{2} \) |
| 79 | \( 1 - 6.57T + 79T^{2} \) |
| 83 | \( 1 - 4.44T + 83T^{2} \) |
| 89 | \( 1 + 4.37T + 89T^{2} \) |
| 97 | \( 1 - 7.51T + 97T^{2} \) |
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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
−8.453006041623811046760756469714, −7.80510604514323467755754260025, −7.17264546118480661108609900862, −6.47482354929847408663249518030, −5.37758593921993309919959396473, −4.10740368848145110878959535987, −3.57622705351919046790424166096, −2.24883966860982533971088291485, −1.06158133095603163744717622414, 0,
1.06158133095603163744717622414, 2.24883966860982533971088291485, 3.57622705351919046790424166096, 4.10740368848145110878959535987, 5.37758593921993309919959396473, 6.47482354929847408663249518030, 7.17264546118480661108609900862, 7.80510604514323467755754260025, 8.453006041623811046760756469714
