L(s) = 1 | − 5-s − 7-s − 6·11-s − 13-s − 6·17-s − 2·19-s − 23-s + 25-s − 9·29-s + 31-s + 35-s + 8·37-s − 9·41-s − 2·43-s − 3·47-s + 49-s + 6·55-s + 12·59-s + 8·61-s + 65-s − 8·67-s + 9·71-s + 5·73-s + 6·77-s − 14·79-s + 6·83-s + 6·85-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.377·7-s − 1.80·11-s − 0.277·13-s − 1.45·17-s − 0.458·19-s − 0.208·23-s + 1/5·25-s − 1.67·29-s + 0.179·31-s + 0.169·35-s + 1.31·37-s − 1.40·41-s − 0.304·43-s − 0.437·47-s + 1/7·49-s + 0.809·55-s + 1.56·59-s + 1.02·61-s + 0.124·65-s − 0.977·67-s + 1.06·71-s + 0.585·73-s + 0.683·77-s − 1.57·79-s + 0.658·83-s + 0.650·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.46094023535042, −13.38537316369736, −12.94352802574298, −12.59338092239280, −11.79760395004162, −11.41223928225117, −10.91363152595391, −10.51909751707363, −9.927845406196946, −9.550681389934870, −8.841151683569304, −8.351657457587389, −7.976419333705176, −7.372583595864030, −6.938135743457024, −6.379846664270124, −5.727771953592111, −5.185971124658781, −4.736185109474990, −4.076515306523316, −3.563338767211411, −2.784095824093164, −2.348473023354065, −1.792488494572024, −0.5388973832348525, 0,
0.5388973832348525, 1.792488494572024, 2.348473023354065, 2.784095824093164, 3.563338767211411, 4.076515306523316, 4.736185109474990, 5.185971124658781, 5.727771953592111, 6.379846664270124, 6.938135743457024, 7.372583595864030, 7.976419333705176, 8.351657457587389, 8.841151683569304, 9.550681389934870, 9.927845406196946, 10.51909751707363, 10.91363152595391, 11.41223928225117, 11.79760395004162, 12.59338092239280, 12.94352802574298, 13.38537316369736, 13.46094023535042