| L(s) = 1 | + 3-s + 9-s + 2·11-s + 13-s + 19-s + 2·23-s + 27-s − 8·29-s + 2·33-s − 7·37-s + 39-s + 2·41-s + 4·43-s + 4·53-s + 57-s + 5·61-s − 67-s + 2·69-s − 7·73-s − 79-s + 81-s − 8·83-s − 8·87-s − 12·89-s − 3·97-s + 2·99-s + 101-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 0.603·11-s + 0.277·13-s + 0.229·19-s + 0.417·23-s + 0.192·27-s − 1.48·29-s + 0.348·33-s − 1.15·37-s + 0.160·39-s + 0.312·41-s + 0.609·43-s + 0.549·53-s + 0.132·57-s + 0.640·61-s − 0.122·67-s + 0.240·69-s − 0.819·73-s − 0.112·79-s + 1/9·81-s − 0.878·83-s − 0.857·87-s − 1.27·89-s − 0.304·97-s + 0.201·99-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.55629360502407, −14.08393071003119, −13.66568090153565, −13.06768982508093, −12.67940844854412, −12.09212235286320, −11.49604594002229, −11.08142777658331, −10.47878782278424, −9.898671690381579, −9.368187492448329, −8.918391176321931, −8.491774773197773, −7.830882028252398, −7.224466628106816, −6.908414452417416, −6.137895381963319, −5.558352306723485, −5.017184103991541, −4.108367653551517, −3.852314330286777, −3.116331389748393, −2.481956952581760, −1.691538174118570, −1.125482746652073, 0,
1.125482746652073, 1.691538174118570, 2.481956952581760, 3.116331389748393, 3.852314330286777, 4.108367653551517, 5.017184103991541, 5.558352306723485, 6.137895381963319, 6.908414452417416, 7.224466628106816, 7.830882028252398, 8.491774773197773, 8.918391176321931, 9.368187492448329, 9.898671690381579, 10.47878782278424, 11.08142777658331, 11.49604594002229, 12.09212235286320, 12.67940844854412, 13.06768982508093, 13.66568090153565, 14.08393071003119, 14.55629360502407
