| L(s) = 1 | + 1.73·2-s + 0.999·4-s + 3·5-s + 3.46·7-s − 1.73·8-s + 5.19·10-s + 1.73·13-s + 5.99·14-s − 5·16-s + 1.73·17-s − 6.92·19-s + 2.99·20-s + 6·23-s + 4·25-s + 2.99·26-s + 3.46·28-s − 1.73·29-s + 4·31-s − 5.19·32-s + 2.99·34-s + 10.3·35-s − 11·37-s − 11.9·38-s − 5.19·40-s − 1.73·41-s − 3.46·43-s + 10.3·46-s + ⋯ |
| L(s) = 1 | + 1.22·2-s + 0.499·4-s + 1.34·5-s + 1.30·7-s − 0.612·8-s + 1.64·10-s + 0.480·13-s + 1.60·14-s − 1.25·16-s + 0.420·17-s − 1.58·19-s + 0.670·20-s + 1.25·23-s + 0.800·25-s + 0.588·26-s + 0.654·28-s − 0.321·29-s + 0.718·31-s − 0.918·32-s + 0.514·34-s + 1.75·35-s − 1.80·37-s − 1.94·38-s − 0.821·40-s − 0.270·41-s − 0.528·43-s + 1.53·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.913867377\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.913867377\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 1.73T + 2T^{2} \) |
| 5 | \( 1 - 3T + 5T^{2} \) |
| 7 | \( 1 - 3.46T + 7T^{2} \) |
| 13 | \( 1 - 1.73T + 13T^{2} \) |
| 17 | \( 1 - 1.73T + 17T^{2} \) |
| 19 | \( 1 + 6.92T + 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 + 1.73T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 11T + 37T^{2} \) |
| 41 | \( 1 + 1.73T + 41T^{2} \) |
| 43 | \( 1 + 3.46T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 9T + 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 2T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 6.92T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 9T + 89T^{2} \) |
| 97 | \( 1 + 7T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.02086063647710924178450116043, −8.870987129926948099475686342112, −8.450993844397007973929666407794, −7.00607633369541907559955104744, −6.19728146879401851309670428527, −5.37454660274404579404395867177, −4.86693552311694469802699538494, −3.83786711455443175585011229911, −2.55800580814670307665353109157, −1.59063782793091227817997843551,
1.59063782793091227817997843551, 2.55800580814670307665353109157, 3.83786711455443175585011229911, 4.86693552311694469802699538494, 5.37454660274404579404395867177, 6.19728146879401851309670428527, 7.00607633369541907559955104744, 8.450993844397007973929666407794, 8.870987129926948099475686342112, 10.02086063647710924178450116043
