| L(s) = 1 | − 2-s + 0.275·3-s + 4-s − 0.829·5-s − 0.275·6-s + 4.83·7-s − 8-s − 2.92·9-s + 0.829·10-s + 11-s + 0.275·12-s − 4.69·13-s − 4.83·14-s − 0.228·15-s + 16-s − 3.61·17-s + 2.92·18-s + 3.96·19-s − 0.829·20-s + 1.33·21-s − 22-s + 3.89·23-s − 0.275·24-s − 4.31·25-s + 4.69·26-s − 1.63·27-s + 4.83·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.159·3-s + 0.5·4-s − 0.370·5-s − 0.112·6-s + 1.82·7-s − 0.353·8-s − 0.974·9-s + 0.262·10-s + 0.301·11-s + 0.0795·12-s − 1.30·13-s − 1.29·14-s − 0.0589·15-s + 0.250·16-s − 0.877·17-s + 0.689·18-s + 0.910·19-s − 0.185·20-s + 0.290·21-s − 0.213·22-s + 0.811·23-s − 0.0562·24-s − 0.862·25-s + 0.920·26-s − 0.314·27-s + 0.914·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 197 | \( 1 - T \) |
| good | 3 | \( 1 - 0.275T + 3T^{2} \) |
| 5 | \( 1 + 0.829T + 5T^{2} \) |
| 7 | \( 1 - 4.83T + 7T^{2} \) |
| 13 | \( 1 + 4.69T + 13T^{2} \) |
| 17 | \( 1 + 3.61T + 17T^{2} \) |
| 19 | \( 1 - 3.96T + 19T^{2} \) |
| 23 | \( 1 - 3.89T + 23T^{2} \) |
| 29 | \( 1 - 7.12T + 29T^{2} \) |
| 31 | \( 1 + 5.82T + 31T^{2} \) |
| 37 | \( 1 + 7.26T + 37T^{2} \) |
| 41 | \( 1 + 2.08T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 - 5.67T + 47T^{2} \) |
| 53 | \( 1 - 8.95T + 53T^{2} \) |
| 59 | \( 1 + 9.73T + 59T^{2} \) |
| 61 | \( 1 + 12.1T + 61T^{2} \) |
| 67 | \( 1 - 12.1T + 67T^{2} \) |
| 71 | \( 1 + 7.84T + 71T^{2} \) |
| 73 | \( 1 + 0.146T + 73T^{2} \) |
| 79 | \( 1 + 6.17T + 79T^{2} \) |
| 83 | \( 1 + 13.7T + 83T^{2} \) |
| 89 | \( 1 + 5.85T + 89T^{2} \) |
| 97 | \( 1 + 12.1T + 97T^{2} \) |
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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
−8.191496749514897533132156303098, −7.36661613410760685026837029408, −6.96946902457276462077582240684, −5.66663154943296024460419124363, −5.07228454914562241369445219547, −4.35359139279048274481240238844, −3.14617587435221737040967999201, −2.26256997950853029206094934864, −1.41905324942858906604982612760, 0,
1.41905324942858906604982612760, 2.26256997950853029206094934864, 3.14617587435221737040967999201, 4.35359139279048274481240238844, 5.07228454914562241369445219547, 5.66663154943296024460419124363, 6.96946902457276462077582240684, 7.36661613410760685026837029408, 8.191496749514897533132156303098
