| L(s) = 1 | + 0.732·3-s − 3.46·5-s + 0.732·7-s − 2.46·9-s + 4.73·11-s + 1.46·13-s − 2.53·15-s − 17-s − 5.46·19-s + 0.535·21-s − 4.73·23-s + 6.99·25-s − 4·27-s + 3.46·29-s − 6.19·31-s + 3.46·33-s − 2.53·35-s − 11.4·37-s + 1.07·39-s − 6·41-s − 12.3·43-s + 8.53·45-s + 6.92·47-s − 6.46·49-s − 0.732·51-s + 0.928·53-s − 16.3·55-s + ⋯ |
| L(s) = 1 | + 0.422·3-s − 1.54·5-s + 0.276·7-s − 0.821·9-s + 1.42·11-s + 0.406·13-s − 0.654·15-s − 0.242·17-s − 1.25·19-s + 0.116·21-s − 0.986·23-s + 1.39·25-s − 0.769·27-s + 0.643·29-s − 1.11·31-s + 0.603·33-s − 0.428·35-s − 1.88·37-s + 0.171·39-s − 0.937·41-s − 1.88·43-s + 1.27·45-s + 1.01·47-s − 0.923·49-s − 0.102·51-s + 0.127·53-s − 2.21·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 0.732T + 3T^{2} \) |
| 5 | \( 1 + 3.46T + 5T^{2} \) |
| 7 | \( 1 - 0.732T + 7T^{2} \) |
| 11 | \( 1 - 4.73T + 11T^{2} \) |
| 13 | \( 1 - 1.46T + 13T^{2} \) |
| 19 | \( 1 + 5.46T + 19T^{2} \) |
| 23 | \( 1 + 4.73T + 23T^{2} \) |
| 29 | \( 1 - 3.46T + 29T^{2} \) |
| 31 | \( 1 + 6.19T + 31T^{2} \) |
| 37 | \( 1 + 11.4T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 12.3T + 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 - 0.928T + 53T^{2} \) |
| 59 | \( 1 + 9.46T + 59T^{2} \) |
| 61 | \( 1 - 7.46T + 61T^{2} \) |
| 67 | \( 1 + 1.07T + 67T^{2} \) |
| 71 | \( 1 - 2.19T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + 1.80T + 79T^{2} \) |
| 83 | \( 1 - 9.46T + 83T^{2} \) |
| 89 | \( 1 - 9.46T + 89T^{2} \) |
| 97 | \( 1 - 8.92T + 97T^{2} \) |
| 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
−9.105116628896838512042105358906, −8.529340441713091781125251981079, −8.066704500498442809580938633487, −6.98774475090211207098656806155, −6.26279449209127142348219497505, −4.91033280036046228394644886774, −3.86375246473730332862260656487, −3.45962131514590313605230875255, −1.85416921761613474096660954846, 0,
1.85416921761613474096660954846, 3.45962131514590313605230875255, 3.86375246473730332862260656487, 4.91033280036046228394644886774, 6.26279449209127142348219497505, 6.98774475090211207098656806155, 8.066704500498442809580938633487, 8.529340441713091781125251981079, 9.105116628896838512042105358906
