| L(s) = 1 | − 3.82·5-s − 0.476·7-s − 11-s + 4.77·13-s + 2.29·17-s + 0.867·19-s + 0.820·23-s + 9.59·25-s − 1.52·29-s + 1.82·31-s + 1.82·35-s − 3.77·37-s − 10.1·41-s − 2.39·43-s + 5.59·47-s − 6.77·49-s − 6.95·53-s + 3.82·55-s − 13.5·59-s + 9.82·61-s − 18.2·65-s − 9.82·67-s + 12.3·71-s − 10.2·73-s + 0.476·77-s − 6.98·79-s − 10.6·83-s + ⋯ |
| L(s) = 1 | − 1.70·5-s − 0.180·7-s − 0.301·11-s + 1.32·13-s + 0.556·17-s + 0.198·19-s + 0.171·23-s + 1.91·25-s − 0.282·29-s + 0.326·31-s + 0.307·35-s − 0.620·37-s − 1.57·41-s − 0.364·43-s + 0.815·47-s − 0.967·49-s − 0.955·53-s + 0.515·55-s − 1.75·59-s + 1.25·61-s − 2.26·65-s − 1.19·67-s + 1.46·71-s − 1.19·73-s + 0.0542·77-s − 0.785·79-s − 1.17·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + 3.82T + 5T^{2} \) |
| 7 | \( 1 + 0.476T + 7T^{2} \) |
| 13 | \( 1 - 4.77T + 13T^{2} \) |
| 17 | \( 1 - 2.29T + 17T^{2} \) |
| 19 | \( 1 - 0.867T + 19T^{2} \) |
| 23 | \( 1 - 0.820T + 23T^{2} \) |
| 29 | \( 1 + 1.52T + 29T^{2} \) |
| 31 | \( 1 - 1.82T + 31T^{2} \) |
| 37 | \( 1 + 3.77T + 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 + 2.39T + 43T^{2} \) |
| 47 | \( 1 - 5.59T + 47T^{2} \) |
| 53 | \( 1 + 6.95T + 53T^{2} \) |
| 59 | \( 1 + 13.5T + 59T^{2} \) |
| 61 | \( 1 - 9.82T + 61T^{2} \) |
| 67 | \( 1 + 9.82T + 67T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 + 6.98T + 79T^{2} \) |
| 83 | \( 1 + 10.6T + 83T^{2} \) |
| 89 | \( 1 - 11.1T + 89T^{2} \) |
| 97 | \( 1 - 5.17T + 97T^{2} \) |
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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
−8.388817235796753340843852673506, −7.947947772723971557850626990425, −7.15702501391858418156981832842, −6.39128624324546302019439973267, −5.34766471034992006784600065196, −4.43652110165151585712784055853, −3.59433377876379816263579762206, −3.08850679422730067224727809645, −1.35155626779663788441515821688, 0,
1.35155626779663788441515821688, 3.08850679422730067224727809645, 3.59433377876379816263579762206, 4.43652110165151585712784055853, 5.34766471034992006784600065196, 6.39128624324546302019439973267, 7.15702501391858418156981832842, 7.947947772723971557850626990425, 8.388817235796753340843852673506
