| L(s) = 1 | + 2-s − 1.57·3-s + 4-s − 2.42·5-s − 1.57·6-s − 1.17·7-s + 8-s − 0.509·9-s − 2.42·10-s + 6.40·11-s − 1.57·12-s + 6.46·13-s − 1.17·14-s + 3.82·15-s + 16-s + 5.17·17-s − 0.509·18-s + 19-s − 2.42·20-s + 1.85·21-s + 6.40·22-s − 3.64·23-s − 1.57·24-s + 0.862·25-s + 6.46·26-s + 5.53·27-s − 1.17·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.911·3-s + 0.5·4-s − 1.08·5-s − 0.644·6-s − 0.443·7-s + 0.353·8-s − 0.169·9-s − 0.765·10-s + 1.93·11-s − 0.455·12-s + 1.79·13-s − 0.313·14-s + 0.986·15-s + 0.250·16-s + 1.25·17-s − 0.120·18-s + 0.229·19-s − 0.541·20-s + 0.403·21-s + 1.36·22-s − 0.760·23-s − 0.322·24-s + 0.172·25-s + 1.26·26-s + 1.06·27-s − 0.221·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.190846155\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.190846155\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 - T \) |
| good | 3 | \( 1 + 1.57T + 3T^{2} \) |
| 5 | \( 1 + 2.42T + 5T^{2} \) |
| 7 | \( 1 + 1.17T + 7T^{2} \) |
| 11 | \( 1 - 6.40T + 11T^{2} \) |
| 13 | \( 1 - 6.46T + 13T^{2} \) |
| 17 | \( 1 - 5.17T + 17T^{2} \) |
| 23 | \( 1 + 3.64T + 23T^{2} \) |
| 29 | \( 1 - 6.13T + 29T^{2} \) |
| 31 | \( 1 + 3.85T + 31T^{2} \) |
| 37 | \( 1 + 7.00T + 37T^{2} \) |
| 41 | \( 1 - 9.28T + 41T^{2} \) |
| 43 | \( 1 + 4.25T + 43T^{2} \) |
| 47 | \( 1 + 0.673T + 47T^{2} \) |
| 53 | \( 1 - 4.28T + 53T^{2} \) |
| 59 | \( 1 - 1.80T + 59T^{2} \) |
| 61 | \( 1 + 8.13T + 61T^{2} \) |
| 67 | \( 1 + 0.187T + 67T^{2} \) |
| 71 | \( 1 - 1.69T + 71T^{2} \) |
| 73 | \( 1 - 12.6T + 73T^{2} \) |
| 79 | \( 1 + 3.46T + 79T^{2} \) |
| 83 | \( 1 + 0.0964T + 83T^{2} \) |
| 89 | \( 1 - 13.6T + 89T^{2} \) |
| 97 | \( 1 - 6.04T + 97T^{2} \) |
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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
−7.72577030327860025430655146309, −6.82313253191042213695608675401, −6.29288530457538708723466383878, −5.92007322296633565740525138412, −5.06887804542597162886401595915, −4.07674766835456810233659349702, −3.70779143556732893783293324418, −3.15610461049273636979545973640, −1.52385105257455141365640279938, −0.75293183431229007196772216781,
0.75293183431229007196772216781, 1.52385105257455141365640279938, 3.15610461049273636979545973640, 3.70779143556732893783293324418, 4.07674766835456810233659349702, 5.06887804542597162886401595915, 5.92007322296633565740525138412, 6.29288530457538708723466383878, 6.82313253191042213695608675401, 7.72577030327860025430655146309
