| L(s) = 1 | + 5-s + 2.37·7-s + 2.37·11-s − 13-s − 4.37·17-s − 4.74·19-s − 6.37·23-s + 25-s + 2·29-s + 4.74·31-s + 2.37·35-s − 0.372·37-s − 4.37·41-s + 4·43-s − 12.7·47-s − 1.37·49-s + 3.62·53-s + 2.37·55-s − 8·59-s − 9.11·61-s − 65-s + 4·67-s − 5.62·71-s + 7.48·73-s + 5.62·77-s − 11.8·79-s − 12·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.896·7-s + 0.715·11-s − 0.277·13-s − 1.06·17-s − 1.08·19-s − 1.32·23-s + 0.200·25-s + 0.371·29-s + 0.852·31-s + 0.400·35-s − 0.0612·37-s − 0.682·41-s + 0.609·43-s − 1.85·47-s − 0.196·49-s + 0.498·53-s + 0.319·55-s − 1.04·59-s − 1.16·61-s − 0.124·65-s + 0.488·67-s − 0.667·71-s + 0.876·73-s + 0.641·77-s − 1.33·79-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 2.37T + 7T^{2} \) |
| 11 | \( 1 - 2.37T + 11T^{2} \) |
| 17 | \( 1 + 4.37T + 17T^{2} \) |
| 19 | \( 1 + 4.74T + 19T^{2} \) |
| 23 | \( 1 + 6.37T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 4.74T + 31T^{2} \) |
| 37 | \( 1 + 0.372T + 37T^{2} \) |
| 41 | \( 1 + 4.37T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 12.7T + 47T^{2} \) |
| 53 | \( 1 - 3.62T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 + 9.11T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 5.62T + 71T^{2} \) |
| 73 | \( 1 - 7.48T + 73T^{2} \) |
| 79 | \( 1 + 11.8T + 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + 7.62T + 89T^{2} \) |
| 97 | \( 1 - 8.37T + 97T^{2} \) |
| 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
−7.37126576520460063186449832926, −6.44517083362422060305877615240, −6.26127953776175934227163672828, −5.21494013140140729895441808495, −4.50725610219306882771222678169, −4.08485578316610031022038986518, −2.91023564875863976948880606652, −2.02770007626584602906840732852, −1.46890181188502866701761087545, 0,
1.46890181188502866701761087545, 2.02770007626584602906840732852, 2.91023564875863976948880606652, 4.08485578316610031022038986518, 4.50725610219306882771222678169, 5.21494013140140729895441808495, 6.26127953776175934227163672828, 6.44517083362422060305877615240, 7.37126576520460063186449832926
