| L(s) = 1 | − 2-s − 1.80·3-s + 4-s + 0.991·5-s + 1.80·6-s + 3.46·7-s − 8-s + 0.263·9-s − 0.991·10-s − 4.83·11-s − 1.80·12-s + 4.44·13-s − 3.46·14-s − 1.79·15-s + 16-s + 4.57·17-s − 0.263·18-s + 5.87·19-s + 0.991·20-s − 6.26·21-s + 4.83·22-s + 0.390·23-s + 1.80·24-s − 4.01·25-s − 4.44·26-s + 4.94·27-s + 3.46·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.04·3-s + 0.5·4-s + 0.443·5-s + 0.737·6-s + 1.31·7-s − 0.353·8-s + 0.0878·9-s − 0.313·10-s − 1.45·11-s − 0.521·12-s + 1.23·13-s − 0.926·14-s − 0.462·15-s + 0.250·16-s + 1.10·17-s − 0.0620·18-s + 1.34·19-s + 0.221·20-s − 1.36·21-s + 1.03·22-s + 0.0813·23-s + 0.368·24-s − 0.803·25-s − 0.871·26-s + 0.951·27-s + 0.655·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.154039040\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.154039040\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2017 | \( 1 - T \) |
| good | 3 | \( 1 + 1.80T + 3T^{2} \) |
| 5 | \( 1 - 0.991T + 5T^{2} \) |
| 7 | \( 1 - 3.46T + 7T^{2} \) |
| 11 | \( 1 + 4.83T + 11T^{2} \) |
| 13 | \( 1 - 4.44T + 13T^{2} \) |
| 17 | \( 1 - 4.57T + 17T^{2} \) |
| 19 | \( 1 - 5.87T + 19T^{2} \) |
| 23 | \( 1 - 0.390T + 23T^{2} \) |
| 29 | \( 1 + 5.36T + 29T^{2} \) |
| 31 | \( 1 + 3.85T + 31T^{2} \) |
| 37 | \( 1 - 1.64T + 37T^{2} \) |
| 41 | \( 1 - 0.615T + 41T^{2} \) |
| 43 | \( 1 - 3.89T + 43T^{2} \) |
| 47 | \( 1 - 9.56T + 47T^{2} \) |
| 53 | \( 1 - 5.41T + 53T^{2} \) |
| 59 | \( 1 - 2.15T + 59T^{2} \) |
| 61 | \( 1 + 0.659T + 61T^{2} \) |
| 67 | \( 1 + 3.75T + 67T^{2} \) |
| 71 | \( 1 - 7.60T + 71T^{2} \) |
| 73 | \( 1 - 7.96T + 73T^{2} \) |
| 79 | \( 1 - 7.32T + 79T^{2} \) |
| 83 | \( 1 + 4.60T + 83T^{2} \) |
| 89 | \( 1 + 17.5T + 89T^{2} \) |
| 97 | \( 1 + 0.900T + 97T^{2} \) |
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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
−8.274747260378705300899376787396, −7.79019496013314061879467648008, −7.18613967373823620960038685756, −5.94914822069828515464989133343, −5.52044453558235702998136346940, −5.17940918120147445181216577891, −3.85540095247260219393361863117, −2.71032963301369247214774760345, −1.63020625287771122530318080821, −0.76275588855155122188253735242,
0.76275588855155122188253735242, 1.63020625287771122530318080821, 2.71032963301369247214774760345, 3.85540095247260219393361863117, 5.17940918120147445181216577891, 5.52044453558235702998136346940, 5.94914822069828515464989133343, 7.18613967373823620960038685756, 7.79019496013314061879467648008, 8.274747260378705300899376787396
