| L(s) = 1 | − 2.64·2-s + 5.00·4-s + 2.64·5-s − 7-s − 7.93·8-s − 7.00·10-s + 2.64·11-s − 2·13-s + 2.64·14-s + 11.0·16-s + 7·19-s + 13.2·20-s − 7.00·22-s + 7.93·23-s + 2.00·25-s + 5.29·26-s − 5.00·28-s − 5.29·29-s + 3·31-s − 13.2·32-s − 2.64·35-s − 3·37-s − 18.5·38-s − 21.0·40-s − 2.64·41-s + 8·43-s + 13.2·44-s + ⋯ |
| L(s) = 1 | − 1.87·2-s + 2.50·4-s + 1.18·5-s − 0.377·7-s − 2.80·8-s − 2.21·10-s + 0.797·11-s − 0.554·13-s + 0.707·14-s + 2.75·16-s + 1.60·19-s + 2.95·20-s − 1.49·22-s + 1.65·23-s + 0.400·25-s + 1.03·26-s − 0.944·28-s − 0.982·29-s + 0.538·31-s − 2.33·32-s − 0.447·35-s − 0.493·37-s − 3.00·38-s − 3.32·40-s − 0.413·41-s + 1.21·43-s + 1.99·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6418816544\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6418816544\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 2 | \( 1 + 2.64T + 2T^{2} \) |
| 5 | \( 1 - 2.64T + 5T^{2} \) |
| 11 | \( 1 - 2.64T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 7T + 19T^{2} \) |
| 23 | \( 1 - 7.93T + 23T^{2} \) |
| 29 | \( 1 + 5.29T + 29T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 + 3T + 37T^{2} \) |
| 41 | \( 1 + 2.64T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 + 2T + 67T^{2} \) |
| 71 | \( 1 + 7.93T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 15.8T + 83T^{2} \) |
| 89 | \( 1 - 18.5T + 89T^{2} \) |
| 97 | \( 1 + 12T + 97T^{2} \) |
| 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
−12.19387318073910906895310642034, −11.22973633454396579993969170855, −10.19859789402338506821667194321, −9.427558516375359699621925509862, −9.025486334051499108478195581285, −7.52651246855958083034618111272, −6.71919753984202071510762151020, −5.58688638814471852323610914395, −2.86589602692304008188598883868, −1.37701167359795407767938570487,
1.37701167359795407767938570487, 2.86589602692304008188598883868, 5.58688638814471852323610914395, 6.71919753984202071510762151020, 7.52651246855958083034618111272, 9.025486334051499108478195581285, 9.427558516375359699621925509862, 10.19859789402338506821667194321, 11.22973633454396579993969170855, 12.19387318073910906895310642034
