| L(s) = 1 | − 2.51·2-s + 0.226·3-s + 4.33·4-s − 1.31·5-s − 0.569·6-s − 1.40·7-s − 5.88·8-s − 2.94·9-s + 3.30·10-s − 1.02·11-s + 0.980·12-s − 5.01·13-s + 3.54·14-s − 0.297·15-s + 6.13·16-s + 17-s + 7.42·18-s − 0.951·19-s − 5.70·20-s − 0.318·21-s + 2.57·22-s − 1.09·23-s − 1.32·24-s − 3.27·25-s + 12.6·26-s − 1.34·27-s − 6.10·28-s + ⋯ |
| L(s) = 1 | − 1.77·2-s + 0.130·3-s + 2.16·4-s − 0.587·5-s − 0.232·6-s − 0.532·7-s − 2.07·8-s − 0.982·9-s + 1.04·10-s − 0.307·11-s + 0.282·12-s − 1.39·13-s + 0.947·14-s − 0.0767·15-s + 1.53·16-s + 0.242·17-s + 1.74·18-s − 0.218·19-s − 1.27·20-s − 0.0694·21-s + 0.548·22-s − 0.227·23-s − 0.271·24-s − 0.654·25-s + 2.47·26-s − 0.258·27-s − 1.15·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.02264883309\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02264883309\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 - T \) |
| 353 | \( 1 + T \) |
| good | 2 | \( 1 + 2.51T + 2T^{2} \) |
| 3 | \( 1 - 0.226T + 3T^{2} \) |
| 5 | \( 1 + 1.31T + 5T^{2} \) |
| 7 | \( 1 + 1.40T + 7T^{2} \) |
| 11 | \( 1 + 1.02T + 11T^{2} \) |
| 13 | \( 1 + 5.01T + 13T^{2} \) |
| 19 | \( 1 + 0.951T + 19T^{2} \) |
| 23 | \( 1 + 1.09T + 23T^{2} \) |
| 29 | \( 1 + 6.92T + 29T^{2} \) |
| 31 | \( 1 - 1.56T + 31T^{2} \) |
| 37 | \( 1 + 4.13T + 37T^{2} \) |
| 41 | \( 1 - 5.99T + 41T^{2} \) |
| 43 | \( 1 - 2.31T + 43T^{2} \) |
| 47 | \( 1 + 3.40T + 47T^{2} \) |
| 53 | \( 1 + 7.12T + 53T^{2} \) |
| 59 | \( 1 + 8.87T + 59T^{2} \) |
| 61 | \( 1 - 5.70T + 61T^{2} \) |
| 67 | \( 1 + 4.65T + 67T^{2} \) |
| 71 | \( 1 + 7.50T + 71T^{2} \) |
| 73 | \( 1 + 16.3T + 73T^{2} \) |
| 79 | \( 1 + 12.2T + 79T^{2} \) |
| 83 | \( 1 + 0.783T + 83T^{2} \) |
| 89 | \( 1 + 6.64T + 89T^{2} \) |
| 97 | \( 1 + 1.92T + 97T^{2} \) |
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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
−8.052823800037639922328681827895, −7.62013619778175841400627514737, −7.09038991010015593986999079604, −6.18480029442248428795240534348, −5.51166671032380755375006550574, −4.36038130160081683452005748515, −3.18615064805040833046218692238, −2.60662779367732017498392818572, −1.67124344043055471544975576429, −0.097613595745883962646240575283,
0.097613595745883962646240575283, 1.67124344043055471544975576429, 2.60662779367732017498392818572, 3.18615064805040833046218692238, 4.36038130160081683452005748515, 5.51166671032380755375006550574, 6.18480029442248428795240534348, 7.09038991010015593986999079604, 7.62013619778175841400627514737, 8.052823800037639922328681827895
