| L(s) = 1 | + 1.73·2-s − 3-s + 0.999·4-s − 3·5-s − 1.73·6-s − 3.46·7-s − 1.73·8-s + 9-s − 5.19·10-s − 0.999·12-s − 1.73·13-s − 5.99·14-s + 3·15-s − 5·16-s + 1.73·17-s + 1.73·18-s + 6.92·19-s − 2.99·20-s + 3.46·21-s − 6·23-s + 1.73·24-s + 4·25-s − 2.99·26-s − 27-s − 3.46·28-s − 1.73·29-s + 5.19·30-s + ⋯ |
| L(s) = 1 | + 1.22·2-s − 0.577·3-s + 0.499·4-s − 1.34·5-s − 0.707·6-s − 1.30·7-s − 0.612·8-s + 0.333·9-s − 1.64·10-s − 0.288·12-s − 0.480·13-s − 1.60·14-s + 0.774·15-s − 1.25·16-s + 0.420·17-s + 0.408·18-s + 1.58·19-s − 0.670·20-s + 0.755·21-s − 1.25·23-s + 0.353·24-s + 0.800·25-s − 0.588·26-s − 0.192·27-s − 0.654·28-s − 0.321·29-s + 0.948·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 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
−11.48613840251284076179398338363, −10.17555670712698730983302805703, −9.279137318937476680564254609479, −7.83888142451841888077368799506, −6.90239902824418094849867412203, −5.92665930083780474844521723709, −4.92362723553219248906923643921, −3.81710189102304683427755985034, −3.14712099992739018433219472729, 0,
3.14712099992739018433219472729, 3.81710189102304683427755985034, 4.92362723553219248906923643921, 5.92665930083780474844521723709, 6.90239902824418094849867412203, 7.83888142451841888077368799506, 9.279137318937476680564254609479, 10.17555670712698730983302805703, 11.48613840251284076179398338363
