| L(s) = 1 | − 2-s + 1.07·3-s + 4-s − 1.51·5-s − 1.07·6-s − 4.18·7-s − 8-s − 1.84·9-s + 1.51·10-s + 2.19·11-s + 1.07·12-s − 7.02·13-s + 4.18·14-s − 1.63·15-s + 16-s − 3.65·17-s + 1.84·18-s − 4.14·19-s − 1.51·20-s − 4.49·21-s − 2.19·22-s − 23-s − 1.07·24-s − 2.69·25-s + 7.02·26-s − 5.20·27-s − 4.18·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.620·3-s + 0.5·4-s − 0.678·5-s − 0.438·6-s − 1.58·7-s − 0.353·8-s − 0.614·9-s + 0.479·10-s + 0.661·11-s + 0.310·12-s − 1.94·13-s + 1.11·14-s − 0.421·15-s + 0.250·16-s − 0.887·17-s + 0.434·18-s − 0.950·19-s − 0.339·20-s − 0.981·21-s − 0.467·22-s − 0.208·23-s − 0.219·24-s − 0.539·25-s + 1.37·26-s − 1.00·27-s − 0.790·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.09971938779\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09971938779\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 - 1.07T + 3T^{2} \) |
| 5 | \( 1 + 1.51T + 5T^{2} \) |
| 7 | \( 1 + 4.18T + 7T^{2} \) |
| 11 | \( 1 - 2.19T + 11T^{2} \) |
| 13 | \( 1 + 7.02T + 13T^{2} \) |
| 17 | \( 1 + 3.65T + 17T^{2} \) |
| 19 | \( 1 + 4.14T + 19T^{2} \) |
| 29 | \( 1 - 1.14T + 29T^{2} \) |
| 31 | \( 1 + 7.85T + 31T^{2} \) |
| 37 | \( 1 + 5.97T + 37T^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 + 4.19T + 43T^{2} \) |
| 47 | \( 1 + 4.95T + 47T^{2} \) |
| 53 | \( 1 + 4.62T + 53T^{2} \) |
| 59 | \( 1 - 3.81T + 59T^{2} \) |
| 61 | \( 1 - 9.54T + 61T^{2} \) |
| 67 | \( 1 - 6.92T + 67T^{2} \) |
| 71 | \( 1 - 8.90T + 71T^{2} \) |
| 73 | \( 1 + 12.9T + 73T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 + 2.52T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 + 11.4T + 97T^{2} \) |
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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
−8.172082817124776791612214174191, −7.37867232103597416516223600069, −6.86405389375143588236493644385, −6.24666498096377623729450991396, −5.29439558994547154877841461278, −4.12981395787508372606752779776, −3.53117684888366691157943596080, −2.65196255843653067379140406954, −2.08654091785191572939497210096, −0.16179218839497191146517319059,
0.16179218839497191146517319059, 2.08654091785191572939497210096, 2.65196255843653067379140406954, 3.53117684888366691157943596080, 4.12981395787508372606752779776, 5.29439558994547154877841461278, 6.24666498096377623729450991396, 6.86405389375143588236493644385, 7.37867232103597416516223600069, 8.172082817124776791612214174191
