L(s) = 1 | − 3·3-s + 4·5-s + 4·7-s + 6·9-s − 6·11-s + 4·13-s − 12·15-s − 4·17-s − 7·19-s − 12·21-s + 6·23-s + 11·25-s − 9·27-s + 6·29-s + 2·31-s + 18·33-s + 16·35-s − 12·39-s − 8·43-s + 24·45-s + 9·47-s + 9·49-s + 12·51-s + 9·53-s − 24·55-s + 21·57-s − 11·59-s + ⋯ |
L(s) = 1 | − 1.73·3-s + 1.78·5-s + 1.51·7-s + 2·9-s − 1.80·11-s + 1.10·13-s − 3.09·15-s − 0.970·17-s − 1.60·19-s − 2.61·21-s + 1.25·23-s + 11/5·25-s − 1.73·27-s + 1.11·29-s + 0.359·31-s + 3.13·33-s + 2.70·35-s − 1.92·39-s − 1.21·43-s + 3.57·45-s + 1.31·47-s + 9/7·49-s + 1.68·51-s + 1.23·53-s − 3.23·55-s + 2.78·57-s − 1.43·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324928 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5077 | \( 1 - T \) |
good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−12.86636063254480, −12.49359320048987, −11.77515535502756, −11.35532509945232, −10.83961544375623, −10.63005120777582, −10.36797787307842, −10.07123684206604, −9.059089574488735, −8.763697111151877, −8.369848388827600, −7.689304351770794, −7.057260560750265, −6.610941970457628, −6.137653519639309, −5.803669974665240, −5.351287265965053, −4.806606803041832, −4.741105508641689, −4.132549514191575, −2.937721306416172, −2.426163570646656, −1.899394295530962, −1.377125980479211, −0.8660027953994616, 0,
0.8660027953994616, 1.377125980479211, 1.899394295530962, 2.426163570646656, 2.937721306416172, 4.132549514191575, 4.741105508641689, 4.806606803041832, 5.351287265965053, 5.803669974665240, 6.137653519639309, 6.610941970457628, 7.057260560750265, 7.689304351770794, 8.369848388827600, 8.763697111151877, 9.059089574488735, 10.07123684206604, 10.36797787307842, 10.63005120777582, 10.83961544375623, 11.35532509945232, 11.77515535502756, 12.49359320048987, 12.86636063254480