| L(s) = 1 | + 1.14·2-s − 0.696·4-s + 4.45·5-s − 4.52·7-s − 3.07·8-s + 5.08·10-s + 2.39·11-s + 4.69·13-s − 5.17·14-s − 2.12·16-s + 0.991·17-s − 1.33·19-s − 3.10·20-s + 2.73·22-s + 23-s + 14.8·25-s + 5.36·26-s + 3.15·28-s + 29-s + 3.21·31-s + 3.73·32-s + 1.13·34-s − 20.1·35-s + 4.71·37-s − 1.52·38-s − 13.7·40-s − 7.59·41-s + ⋯ |
| L(s) = 1 | + 0.807·2-s − 0.348·4-s + 1.99·5-s − 1.71·7-s − 1.08·8-s + 1.60·10-s + 0.721·11-s + 1.30·13-s − 1.38·14-s − 0.530·16-s + 0.240·17-s − 0.306·19-s − 0.693·20-s + 0.582·22-s + 0.208·23-s + 2.97·25-s + 1.05·26-s + 0.595·28-s + 0.185·29-s + 0.576·31-s + 0.659·32-s + 0.194·34-s − 3.41·35-s + 0.774·37-s − 0.247·38-s − 2.16·40-s − 1.18·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.372872676\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.372872676\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 1.14T + 2T^{2} \) |
| 5 | \( 1 - 4.45T + 5T^{2} \) |
| 7 | \( 1 + 4.52T + 7T^{2} \) |
| 11 | \( 1 - 2.39T + 11T^{2} \) |
| 13 | \( 1 - 4.69T + 13T^{2} \) |
| 17 | \( 1 - 0.991T + 17T^{2} \) |
| 19 | \( 1 + 1.33T + 19T^{2} \) |
| 31 | \( 1 - 3.21T + 31T^{2} \) |
| 37 | \( 1 - 4.71T + 37T^{2} \) |
| 41 | \( 1 + 7.59T + 41T^{2} \) |
| 43 | \( 1 + 8.56T + 43T^{2} \) |
| 47 | \( 1 + 0.810T + 47T^{2} \) |
| 53 | \( 1 + 6.18T + 53T^{2} \) |
| 59 | \( 1 + 8.95T + 59T^{2} \) |
| 61 | \( 1 + 1.56T + 61T^{2} \) |
| 67 | \( 1 - 6.18T + 67T^{2} \) |
| 71 | \( 1 - 15.1T + 71T^{2} \) |
| 73 | \( 1 - 5.61T + 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 - 1.73T + 83T^{2} \) |
| 89 | \( 1 - 14.5T + 89T^{2} \) |
| 97 | \( 1 - 8.70T + 97T^{2} \) |
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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
−8.355920206236495262167544841478, −6.71394937538443503693417319239, −6.42021708326150949741202492878, −6.05527473658749550644837552373, −5.35319713472382382350243148694, −4.52243361590253438907417899470, −3.40690239346642464379328711510, −3.13681544964340777949798959164, −2.00558619250340789660390041345, −0.874773419823097028254651712328,
0.874773419823097028254651712328, 2.00558619250340789660390041345, 3.13681544964340777949798959164, 3.40690239346642464379328711510, 4.52243361590253438907417899470, 5.35319713472382382350243148694, 6.05527473658749550644837552373, 6.42021708326150949741202492878, 6.71394937538443503693417319239, 8.355920206236495262167544841478
