| L(s) = 1 | − 2.31·2-s + 3.30·3-s + 3.34·4-s + 5-s − 7.64·6-s + 0.225·7-s − 3.09·8-s + 7.95·9-s − 2.31·10-s + 0.662·11-s + 11.0·12-s + 2.92·13-s − 0.521·14-s + 3.30·15-s + 0.481·16-s − 5.12·17-s − 18.3·18-s + 3.17·19-s + 3.34·20-s + 0.746·21-s − 1.53·22-s − 4.47·23-s − 10.2·24-s + 25-s − 6.75·26-s + 16.3·27-s + 0.753·28-s + ⋯ |
| L(s) = 1 | − 1.63·2-s + 1.91·3-s + 1.67·4-s + 0.447·5-s − 3.12·6-s + 0.0852·7-s − 1.09·8-s + 2.65·9-s − 0.730·10-s + 0.199·11-s + 3.19·12-s + 0.810·13-s − 0.139·14-s + 0.854·15-s + 0.120·16-s − 1.24·17-s − 4.33·18-s + 0.728·19-s + 0.747·20-s + 0.162·21-s − 0.326·22-s − 0.933·23-s − 2.09·24-s + 0.200·25-s − 1.32·26-s + 3.15·27-s + 0.142·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.504112457\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.504112457\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 1607 | \( 1 + T \) |
| good | 2 | \( 1 + 2.31T + 2T^{2} \) |
| 3 | \( 1 - 3.30T + 3T^{2} \) |
| 7 | \( 1 - 0.225T + 7T^{2} \) |
| 11 | \( 1 - 0.662T + 11T^{2} \) |
| 13 | \( 1 - 2.92T + 13T^{2} \) |
| 17 | \( 1 + 5.12T + 17T^{2} \) |
| 19 | \( 1 - 3.17T + 19T^{2} \) |
| 23 | \( 1 + 4.47T + 23T^{2} \) |
| 29 | \( 1 - 7.05T + 29T^{2} \) |
| 31 | \( 1 - 6.45T + 31T^{2} \) |
| 37 | \( 1 + 8.74T + 37T^{2} \) |
| 41 | \( 1 - 7.33T + 41T^{2} \) |
| 43 | \( 1 + 3.51T + 43T^{2} \) |
| 47 | \( 1 + 1.72T + 47T^{2} \) |
| 53 | \( 1 - 1.12T + 53T^{2} \) |
| 59 | \( 1 - 4.26T + 59T^{2} \) |
| 61 | \( 1 - 2.82T + 61T^{2} \) |
| 67 | \( 1 - 9.95T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 + 14.9T + 73T^{2} \) |
| 79 | \( 1 + 9.47T + 79T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 - 4.22T + 89T^{2} \) |
| 97 | \( 1 + 5.40T + 97T^{2} \) |
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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
−8.164509434436399557150181255173, −7.50202798875163490117735524594, −6.75688577237049211506794616445, −6.33095128101998744144245877518, −4.83534564643301421002499788332, −4.02091585392189526238140659423, −3.12729612963742523583871295142, −2.36826657935054808519711724067, −1.77365017746050213856729583520, −0.965817941739383043150425081016,
0.965817941739383043150425081016, 1.77365017746050213856729583520, 2.36826657935054808519711724067, 3.12729612963742523583871295142, 4.02091585392189526238140659423, 4.83534564643301421002499788332, 6.33095128101998744144245877518, 6.75688577237049211506794616445, 7.50202798875163490117735524594, 8.164509434436399557150181255173
