| L(s) = 1 | + 2-s + 0.414·3-s + 4-s + 0.0594·5-s + 0.414·6-s − 4.22·7-s + 8-s − 2.82·9-s + 0.0594·10-s + 2.16·11-s + 0.414·12-s − 3.13·13-s − 4.22·14-s + 0.0246·15-s + 16-s − 4.80·17-s − 2.82·18-s − 1.91·19-s + 0.0594·20-s − 1.74·21-s + 2.16·22-s + 23-s + 0.414·24-s − 4.99·25-s − 3.13·26-s − 2.41·27-s − 4.22·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.239·3-s + 0.5·4-s + 0.0265·5-s + 0.169·6-s − 1.59·7-s + 0.353·8-s − 0.942·9-s + 0.0187·10-s + 0.652·11-s + 0.119·12-s − 0.870·13-s − 1.12·14-s + 0.00635·15-s + 0.250·16-s − 1.16·17-s − 0.666·18-s − 0.439·19-s + 0.0132·20-s − 0.381·21-s + 0.461·22-s + 0.208·23-s + 0.0845·24-s − 0.999·25-s − 0.615·26-s − 0.464·27-s − 0.798·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - 0.414T + 3T^{2} \) |
| 5 | \( 1 - 0.0594T + 5T^{2} \) |
| 7 | \( 1 + 4.22T + 7T^{2} \) |
| 11 | \( 1 - 2.16T + 11T^{2} \) |
| 13 | \( 1 + 3.13T + 13T^{2} \) |
| 17 | \( 1 + 4.80T + 17T^{2} \) |
| 19 | \( 1 + 1.91T + 19T^{2} \) |
| 31 | \( 1 - 3.11T + 31T^{2} \) |
| 37 | \( 1 - 5.83T + 37T^{2} \) |
| 41 | \( 1 + 8.38T + 41T^{2} \) |
| 43 | \( 1 + 0.123T + 43T^{2} \) |
| 47 | \( 1 + 4.50T + 47T^{2} \) |
| 53 | \( 1 - 1.23T + 53T^{2} \) |
| 59 | \( 1 + 5.31T + 59T^{2} \) |
| 61 | \( 1 + 8.39T + 61T^{2} \) |
| 67 | \( 1 - 5.55T + 67T^{2} \) |
| 71 | \( 1 + 5.10T + 71T^{2} \) |
| 73 | \( 1 - 8.64T + 73T^{2} \) |
| 79 | \( 1 + 3.37T + 79T^{2} \) |
| 83 | \( 1 + 0.186T + 83T^{2} \) |
| 89 | \( 1 - 14.8T + 89T^{2} \) |
| 97 | \( 1 - 3.26T + 97T^{2} \) |
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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
−9.332561988718871786579504275129, −8.480896622976677983085288219193, −7.38104752528184804869613891605, −6.44682681657032738807766712660, −6.12153387314983760478548074890, −4.91954562048224095651166034682, −3.89345538412091548834274852479, −3.06083417629139340583072236360, −2.22021489681909615341332417645, 0,
2.22021489681909615341332417645, 3.06083417629139340583072236360, 3.89345538412091548834274852479, 4.91954562048224095651166034682, 6.12153387314983760478548074890, 6.44682681657032738807766712660, 7.38104752528184804869613891605, 8.480896622976677983085288219193, 9.332561988718871786579504275129
