| L(s) = 1 | + 2·3-s − 5-s − 7-s + 9-s − 4·11-s − 2·15-s − 6·19-s − 2·21-s + 2·23-s + 25-s − 4·27-s + 6·29-s − 8·31-s − 8·33-s + 35-s + 6·37-s + 8·41-s − 4·43-s − 45-s − 8·47-s + 49-s + 4·55-s − 12·57-s − 10·59-s − 14·61-s − 63-s − 4·67-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.20·11-s − 0.516·15-s − 1.37·19-s − 0.436·21-s + 0.417·23-s + 1/5·25-s − 0.769·27-s + 1.11·29-s − 1.43·31-s − 1.39·33-s + 0.169·35-s + 0.986·37-s + 1.24·41-s − 0.609·43-s − 0.149·45-s − 1.16·47-s + 1/7·49-s + 0.539·55-s − 1.58·57-s − 1.30·59-s − 1.79·61-s − 0.125·63-s − 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 94640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 94640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9229838709\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9229838709\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
| 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
−13.75062246910754, −13.33480074280829, −12.83495916269832, −12.63640032117309, −11.93308219122837, −11.20778326706886, −10.83347781308144, −10.37867429411190, −9.754093649162326, −9.227901937903378, −8.778613990584142, −8.293871028641609, −7.864442295267999, −7.437004034084287, −6.875669158629411, −6.076673168328811, −5.778182063187481, −4.747124473512103, −4.530668653599197, −3.730876456458763, −3.132354330896935, −2.760540475550529, −2.187161679298744, −1.440259591625480, −0.2666206692220682,
0.2666206692220682, 1.440259591625480, 2.187161679298744, 2.760540475550529, 3.132354330896935, 3.730876456458763, 4.530668653599197, 4.747124473512103, 5.778182063187481, 6.076673168328811, 6.875669158629411, 7.437004034084287, 7.864442295267999, 8.293871028641609, 8.778613990584142, 9.227901937903378, 9.754093649162326, 10.37867429411190, 10.83347781308144, 11.20778326706886, 11.93308219122837, 12.63640032117309, 12.83495916269832, 13.33480074280829, 13.75062246910754
