| L(s) = 1 | + 3-s + 2·7-s + 9-s − 11-s − 2·13-s − 2·19-s + 2·21-s + 27-s − 8·31-s − 33-s − 2·37-s − 2·39-s + 2·43-s − 3·49-s − 6·53-s − 2·57-s + 12·59-s + 2·61-s + 2·63-s − 4·67-s − 2·73-s − 2·77-s + 10·79-s + 81-s − 12·83-s − 6·89-s − 4·91-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.755·7-s + 1/3·9-s − 0.301·11-s − 0.554·13-s − 0.458·19-s + 0.436·21-s + 0.192·27-s − 1.43·31-s − 0.174·33-s − 0.328·37-s − 0.320·39-s + 0.304·43-s − 3/7·49-s − 0.824·53-s − 0.264·57-s + 1.56·59-s + 0.256·61-s + 0.251·63-s − 0.488·67-s − 0.234·73-s − 0.227·77-s + 1.12·79-s + 1/9·81-s − 1.31·83-s − 0.635·89-s − 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−16.31866343371476, −16.05904928478222, −15.19219188598535, −14.74538456743085, −14.45860448815758, −13.73038134049211, −13.20453856327660, −12.54608456038314, −12.13393353084803, −11.18190039406965, −10.96552438182501, −10.11449191134802, −9.610359423347394, −8.911159346897237, −8.354605300610627, −7.806422153784684, −7.216582565620551, −6.608388921926662, −5.622144788988497, −5.109476349886424, −4.355005572759359, −3.712024739615051, −2.810956266226967, −2.112133730398711, −1.369082201209844, 0,
1.369082201209844, 2.112133730398711, 2.810956266226967, 3.712024739615051, 4.355005572759359, 5.109476349886424, 5.622144788988497, 6.608388921926662, 7.216582565620551, 7.806422153784684, 8.354605300610627, 8.911159346897237, 9.610359423347394, 10.11449191134802, 10.96552438182501, 11.18190039406965, 12.13393353084803, 12.54608456038314, 13.20453856327660, 13.73038134049211, 14.45860448815758, 14.74538456743085, 15.19219188598535, 16.05904928478222, 16.31866343371476
