| L(s) = 1 | + 1.58·2-s − 1.11·3-s + 0.500·4-s + 3.47·5-s − 1.76·6-s − 2.37·8-s − 1.75·9-s + 5.49·10-s − 5.55·11-s − 0.557·12-s + 5.94·13-s − 3.87·15-s − 4.75·16-s − 1.11·17-s − 2.77·18-s − 7.21·19-s + 1.74·20-s − 8.79·22-s + 3.36·23-s + 2.64·24-s + 7.09·25-s + 9.40·26-s + 5.30·27-s + 5.92·29-s − 6.13·30-s − 0.933·31-s − 2.76·32-s + ⋯ |
| L(s) = 1 | + 1.11·2-s − 0.643·3-s + 0.250·4-s + 1.55·5-s − 0.719·6-s − 0.838·8-s − 0.585·9-s + 1.73·10-s − 1.67·11-s − 0.161·12-s + 1.64·13-s − 1.00·15-s − 1.18·16-s − 0.271·17-s − 0.654·18-s − 1.65·19-s + 0.389·20-s − 1.87·22-s + 0.702·23-s + 0.539·24-s + 1.41·25-s + 1.84·26-s + 1.02·27-s + 1.10·29-s − 1.11·30-s − 0.167·31-s − 0.489·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.798059799\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.798059799\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 - 1.58T + 2T^{2} \) |
| 3 | \( 1 + 1.11T + 3T^{2} \) |
| 5 | \( 1 - 3.47T + 5T^{2} \) |
| 11 | \( 1 + 5.55T + 11T^{2} \) |
| 13 | \( 1 - 5.94T + 13T^{2} \) |
| 17 | \( 1 + 1.11T + 17T^{2} \) |
| 19 | \( 1 + 7.21T + 19T^{2} \) |
| 23 | \( 1 - 3.36T + 23T^{2} \) |
| 29 | \( 1 - 5.92T + 29T^{2} \) |
| 31 | \( 1 + 0.933T + 31T^{2} \) |
| 37 | \( 1 - 0.869T + 37T^{2} \) |
| 41 | \( 1 - 1.05T + 41T^{2} \) |
| 43 | \( 1 - 4.96T + 43T^{2} \) |
| 47 | \( 1 - 12.2T + 47T^{2} \) |
| 53 | \( 1 + 0.972T + 53T^{2} \) |
| 59 | \( 1 + 1.21T + 59T^{2} \) |
| 61 | \( 1 - 2.39T + 61T^{2} \) |
| 67 | \( 1 - 5.12T + 67T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 - 8.77T + 73T^{2} \) |
| 79 | \( 1 - 5.63T + 79T^{2} \) |
| 83 | \( 1 + 11.2T + 83T^{2} \) |
| 89 | \( 1 - 14.3T + 89T^{2} \) |
| 97 | \( 1 - 10.0T + 97T^{2} \) |
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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
−8.248843840881232001712966123023, −6.84304625838439130741086680257, −6.09890489248387694116376151535, −5.97899997003550696772596869151, −5.22856537554947594815383225081, −4.74836113378871118713523241085, −3.72872920642329941496599094995, −2.68192548068808400982834170450, −2.26611255586144597876983924385, −0.73852942459426958076838765930,
0.73852942459426958076838765930, 2.26611255586144597876983924385, 2.68192548068808400982834170450, 3.72872920642329941496599094995, 4.74836113378871118713523241085, 5.22856537554947594815383225081, 5.97899997003550696772596869151, 6.09890489248387694116376151535, 6.84304625838439130741086680257, 8.248843840881232001712966123023
