| L(s) = 1 | − 3-s − 4·7-s + 9-s + 11-s − 13-s + 4·19-s + 4·21-s − 5·25-s − 27-s − 10·31-s − 33-s − 2·37-s + 39-s − 6·41-s + 10·43-s + 9·49-s − 6·53-s − 4·57-s − 2·61-s − 4·63-s − 2·67-s + 12·71-s − 10·73-s + 5·75-s − 4·77-s − 10·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s + 0.301·11-s − 0.277·13-s + 0.917·19-s + 0.872·21-s − 25-s − 0.192·27-s − 1.79·31-s − 0.174·33-s − 0.328·37-s + 0.160·39-s − 0.937·41-s + 1.52·43-s + 9/7·49-s − 0.824·53-s − 0.529·57-s − 0.256·61-s − 0.503·63-s − 0.244·67-s + 1.42·71-s − 1.17·73-s + 0.577·75-s − 0.455·77-s − 1.12·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6338519273\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6338519273\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−15.39917355742381, −14.79297498361681, −14.03402341725504, −13.67513978763697, −12.90961635227631, −12.67249636703880, −12.04935727432683, −11.57329937477247, −10.94316880964422, −10.29796653669728, −9.859027307691225, −9.203044210749516, −9.060441376465259, −7.869530140544209, −7.450232414224498, −6.783803235413204, −6.348550198902870, −5.642051998328849, −5.289965245292405, −4.310493199591030, −3.663468282098463, −3.188854303866913, −2.290623571291254, −1.387496831875822, −0.3273778286482822,
0.3273778286482822, 1.387496831875822, 2.290623571291254, 3.188854303866913, 3.663468282098463, 4.310493199591030, 5.289965245292405, 5.642051998328849, 6.348550198902870, 6.783803235413204, 7.450232414224498, 7.869530140544209, 9.060441376465259, 9.203044210749516, 9.859027307691225, 10.29796653669728, 10.94316880964422, 11.57329937477247, 12.04935727432683, 12.67249636703880, 12.90961635227631, 13.67513978763697, 14.03402341725504, 14.79297498361681, 15.39917355742381
