| L(s) = 1 | + 0.414·3-s − 3.82·5-s − 4·7-s − 2.82·9-s − 2.41·11-s + 4.65·13-s − 1.58·15-s + 3.65·17-s + 2·19-s − 1.65·21-s − 4.82·23-s + 9.65·25-s − 2.41·27-s + 29-s − 8.41·31-s − 0.999·33-s + 15.3·35-s − 1.65·37-s + 1.92·39-s − 9.65·41-s + 1.58·43-s + 10.8·45-s − 12.0·47-s + 9·49-s + 1.51·51-s − 7·53-s + 9.24·55-s + ⋯ |
| L(s) = 1 | + 0.239·3-s − 1.71·5-s − 1.51·7-s − 0.942·9-s − 0.727·11-s + 1.29·13-s − 0.409·15-s + 0.886·17-s + 0.458·19-s − 0.361·21-s − 1.00·23-s + 1.93·25-s − 0.464·27-s + 0.185·29-s − 1.51·31-s − 0.174·33-s + 2.58·35-s − 0.272·37-s + 0.308·39-s − 1.50·41-s + 0.241·43-s + 1.61·45-s − 1.76·47-s + 1.28·49-s + 0.212·51-s − 0.961·53-s + 1.24·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 0.414T + 3T^{2} \) |
| 5 | \( 1 + 3.82T + 5T^{2} \) |
| 7 | \( 1 + 4T + 7T^{2} \) |
| 11 | \( 1 + 2.41T + 11T^{2} \) |
| 13 | \( 1 - 4.65T + 13T^{2} \) |
| 17 | \( 1 - 3.65T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 4.82T + 23T^{2} \) |
| 31 | \( 1 + 8.41T + 31T^{2} \) |
| 37 | \( 1 + 1.65T + 37T^{2} \) |
| 41 | \( 1 + 9.65T + 41T^{2} \) |
| 43 | \( 1 - 1.58T + 43T^{2} \) |
| 47 | \( 1 + 12.0T + 47T^{2} \) |
| 53 | \( 1 + 7T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 5.65T + 67T^{2} \) |
| 71 | \( 1 - 6.48T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 - 9.72T + 79T^{2} \) |
| 83 | \( 1 + 8.82T + 83T^{2} \) |
| 89 | \( 1 + 9.65T + 89T^{2} \) |
| 97 | \( 1 + 7.31T + 97T^{2} \) |
| 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
−11.73538974419154243459943507228, −10.90643215754253071387027061070, −9.751523384666126644758475364707, −8.529442525447824558118698080308, −7.905704047588100072501880577631, −6.72730713926566017292704934627, −5.51414059763025832972764105195, −3.67453541841993081077723014181, −3.23564096215026673715329578915, 0,
3.23564096215026673715329578915, 3.67453541841993081077723014181, 5.51414059763025832972764105195, 6.72730713926566017292704934627, 7.905704047588100072501880577631, 8.529442525447824558118698080308, 9.751523384666126644758475364707, 10.90643215754253071387027061070, 11.73538974419154243459943507228
