| L(s) = 1 | − 2·5-s − 3·13-s − 4·17-s − 4·19-s + 2·23-s − 25-s − 4·29-s − 31-s − 3·37-s + 8·41-s − 43-s − 8·47-s + 14·53-s − 12·59-s − 61-s + 6·65-s + 3·67-s − 6·71-s − 10·73-s + 79-s + 6·83-s + 8·85-s + 2·89-s + 8·95-s − 97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.832·13-s − 0.970·17-s − 0.917·19-s + 0.417·23-s − 1/5·25-s − 0.742·29-s − 0.179·31-s − 0.493·37-s + 1.24·41-s − 0.152·43-s − 1.16·47-s + 1.92·53-s − 1.56·59-s − 0.128·61-s + 0.744·65-s + 0.366·67-s − 0.712·71-s − 1.17·73-s + 0.112·79-s + 0.658·83-s + 0.867·85-s + 0.211·89-s + 0.820·95-s − 0.101·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7575027595\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7575027595\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
| 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
−16.50496959491817, −15.99349313657811, −15.27188815560882, −15.00207425663397, −14.44459341769585, −13.62060014993086, −13.04560596202993, −12.53537786538580, −11.87871997495172, −11.38838872512363, −10.79291342353724, −10.23285832239211, −9.376514104531715, −8.889462791540824, −8.209294289495245, −7.540400922256451, −7.077733847270682, −6.340702897000968, −5.564823793753000, −4.678540060021651, −4.238922856091131, −3.467716689956600, −2.566731334143596, −1.790905563573331, −0.3897699854692814,
0.3897699854692814, 1.790905563573331, 2.566731334143596, 3.467716689956600, 4.238922856091131, 4.678540060021651, 5.564823793753000, 6.340702897000968, 7.077733847270682, 7.540400922256451, 8.209294289495245, 8.889462791540824, 9.376514104531715, 10.23285832239211, 10.79291342353724, 11.38838872512363, 11.87871997495172, 12.53537786538580, 13.04560596202993, 13.62060014993086, 14.44459341769585, 15.00207425663397, 15.27188815560882, 15.99349313657811, 16.50496959491817
