| L(s) = 1 | − 5-s + 4.77·7-s − 3.77·11-s + 3·13-s + 1.77·17-s − 2.77·19-s + 7.77·23-s + 25-s + 3.77·29-s + 2.22·31-s − 4.77·35-s − 0.772·37-s − 9.54·41-s − 1.77·43-s + 5.77·47-s + 15.7·49-s − 9.54·53-s + 3.77·55-s + 12·59-s + 0.772·61-s − 3·65-s + 14.7·67-s + 13.5·71-s + 4.77·73-s − 18·77-s − 5·79-s + 6·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.80·7-s − 1.13·11-s + 0.832·13-s + 0.429·17-s − 0.635·19-s + 1.62·23-s + 0.200·25-s + 0.700·29-s + 0.400·31-s − 0.806·35-s − 0.126·37-s − 1.49·41-s − 0.270·43-s + 0.841·47-s + 2.25·49-s − 1.31·53-s + 0.508·55-s + 1.56·59-s + 0.0988·61-s − 0.372·65-s + 1.80·67-s + 1.60·71-s + 0.558·73-s − 2.05·77-s − 0.562·79-s + 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.820747334\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.820747334\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| good | 7 | \( 1 - 4.77T + 7T^{2} \) |
| 11 | \( 1 + 3.77T + 11T^{2} \) |
| 13 | \( 1 - 3T + 13T^{2} \) |
| 17 | \( 1 - 1.77T + 17T^{2} \) |
| 19 | \( 1 + 2.77T + 19T^{2} \) |
| 23 | \( 1 - 7.77T + 23T^{2} \) |
| 29 | \( 1 - 3.77T + 29T^{2} \) |
| 31 | \( 1 - 2.22T + 31T^{2} \) |
| 37 | \( 1 + 0.772T + 37T^{2} \) |
| 41 | \( 1 + 9.54T + 41T^{2} \) |
| 43 | \( 1 + 1.77T + 43T^{2} \) |
| 47 | \( 1 - 5.77T + 47T^{2} \) |
| 53 | \( 1 + 9.54T + 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 0.772T + 61T^{2} \) |
| 67 | \( 1 - 14.7T + 67T^{2} \) |
| 71 | \( 1 - 13.5T + 71T^{2} \) |
| 73 | \( 1 - 4.77T + 73T^{2} \) |
| 79 | \( 1 + 5T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 - 8T + 89T^{2} \) |
| 97 | \( 1 + 18.3T + 97T^{2} \) |
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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
−10.05376734710475557817047260767, −8.656145276158986512625818520066, −8.310760603586454466405552170874, −7.59110103085162998070285229428, −6.61443514882391073363660120098, −5.23900535721514374115619062340, −4.89501505238091724566000637592, −3.70637045991134894000705535622, −2.42458889797518145421757105454, −1.12376063371048189774641498016,
1.12376063371048189774641498016, 2.42458889797518145421757105454, 3.70637045991134894000705535622, 4.89501505238091724566000637592, 5.23900535721514374115619062340, 6.61443514882391073363660120098, 7.59110103085162998070285229428, 8.310760603586454466405552170874, 8.656145276158986512625818520066, 10.05376734710475557817047260767
