| L(s) = 1 | − 2-s + 1.11·3-s + 4-s − 0.821·5-s − 1.11·6-s − 1.96·7-s − 8-s − 1.76·9-s + 0.821·10-s − 1.01·11-s + 1.11·12-s + 3.29·13-s + 1.96·14-s − 0.913·15-s + 16-s + 6.58·17-s + 1.76·18-s − 1.47·19-s − 0.821·20-s − 2.18·21-s + 1.01·22-s − 5.87·23-s − 1.11·24-s − 4.32·25-s − 3.29·26-s − 5.29·27-s − 1.96·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.642·3-s + 0.5·4-s − 0.367·5-s − 0.454·6-s − 0.741·7-s − 0.353·8-s − 0.587·9-s + 0.259·10-s − 0.306·11-s + 0.321·12-s + 0.913·13-s + 0.523·14-s − 0.235·15-s + 0.250·16-s + 1.59·17-s + 0.415·18-s − 0.337·19-s − 0.183·20-s − 0.476·21-s + 0.216·22-s − 1.22·23-s − 0.227·24-s − 0.865·25-s − 0.645·26-s − 1.01·27-s − 0.370·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 2011 | \( 1 + T \) |
| good | 3 | \( 1 - 1.11T + 3T^{2} \) |
| 5 | \( 1 + 0.821T + 5T^{2} \) |
| 7 | \( 1 + 1.96T + 7T^{2} \) |
| 11 | \( 1 + 1.01T + 11T^{2} \) |
| 13 | \( 1 - 3.29T + 13T^{2} \) |
| 17 | \( 1 - 6.58T + 17T^{2} \) |
| 19 | \( 1 + 1.47T + 19T^{2} \) |
| 23 | \( 1 + 5.87T + 23T^{2} \) |
| 29 | \( 1 - 10.0T + 29T^{2} \) |
| 31 | \( 1 - 7.85T + 31T^{2} \) |
| 37 | \( 1 + 0.0730T + 37T^{2} \) |
| 41 | \( 1 + 4.21T + 41T^{2} \) |
| 43 | \( 1 + 0.336T + 43T^{2} \) |
| 47 | \( 1 + 7.93T + 47T^{2} \) |
| 53 | \( 1 + 8.04T + 53T^{2} \) |
| 59 | \( 1 - 2.93T + 59T^{2} \) |
| 61 | \( 1 + 9.34T + 61T^{2} \) |
| 67 | \( 1 - 13.8T + 67T^{2} \) |
| 71 | \( 1 - 14.5T + 71T^{2} \) |
| 73 | \( 1 + 7.84T + 73T^{2} \) |
| 79 | \( 1 + 5.15T + 79T^{2} \) |
| 83 | \( 1 + 4.36T + 83T^{2} \) |
| 89 | \( 1 - 14.4T + 89T^{2} \) |
| 97 | \( 1 + 11.3T + 97T^{2} \) |
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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
−8.142121800932024031370358327951, −7.80354610694856545101415601152, −6.51093841854161473684450662674, −6.19553846863187580403745014616, −5.19340816056594621457018817760, −3.91556736343780401066794860085, −3.25014928320170520829771650913, −2.58440215900839253680719650851, −1.32467349787044723730558127553, 0,
1.32467349787044723730558127553, 2.58440215900839253680719650851, 3.25014928320170520829771650913, 3.91556736343780401066794860085, 5.19340816056594621457018817760, 6.19553846863187580403745014616, 6.51093841854161473684450662674, 7.80354610694856545101415601152, 8.142121800932024031370358327951
