| L(s) = 1 | + 2-s + 4-s + 2·5-s − 1.56·7-s + 8-s + 2·10-s − 1.56·11-s + 6.68·13-s − 1.56·14-s + 16-s + 3.56·17-s − 3.56·19-s + 2·20-s − 1.56·22-s + 8.68·23-s − 25-s + 6.68·26-s − 1.56·28-s − 1.12·29-s − 9.12·31-s + 32-s + 3.56·34-s − 3.12·35-s − 37-s − 3.56·38-s + 2·40-s + 11.1·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.894·5-s − 0.590·7-s + 0.353·8-s + 0.632·10-s − 0.470·11-s + 1.85·13-s − 0.417·14-s + 0.250·16-s + 0.863·17-s − 0.817·19-s + 0.447·20-s − 0.332·22-s + 1.81·23-s − 0.200·25-s + 1.31·26-s − 0.295·28-s − 0.208·29-s − 1.63·31-s + 0.176·32-s + 0.610·34-s − 0.527·35-s − 0.164·37-s − 0.577·38-s + 0.316·40-s + 1.73·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.623494085\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.623494085\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 5 | \( 1 - 2T + 5T^{2} \) |
| 7 | \( 1 + 1.56T + 7T^{2} \) |
| 11 | \( 1 + 1.56T + 11T^{2} \) |
| 13 | \( 1 - 6.68T + 13T^{2} \) |
| 17 | \( 1 - 3.56T + 17T^{2} \) |
| 19 | \( 1 + 3.56T + 19T^{2} \) |
| 23 | \( 1 - 8.68T + 23T^{2} \) |
| 29 | \( 1 + 1.12T + 29T^{2} \) |
| 31 | \( 1 + 9.12T + 31T^{2} \) |
| 41 | \( 1 - 11.1T + 41T^{2} \) |
| 43 | \( 1 - 1.12T + 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 + 1.56T + 53T^{2} \) |
| 59 | \( 1 - 0.876T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 - 2.24T + 67T^{2} \) |
| 71 | \( 1 - 2.24T + 71T^{2} \) |
| 73 | \( 1 - 3.56T + 73T^{2} \) |
| 79 | \( 1 + 6T + 79T^{2} \) |
| 83 | \( 1 - 14.9T + 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 - 2.87T + 97T^{2} \) |
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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
−10.82492279477294017339874254148, −9.666111955655835904257090591884, −8.921140041973633294089213449529, −7.79780882030356989594681079953, −6.63652080618383635307639760474, −5.96544996271596631177230735419, −5.22921359046827082919555720214, −3.84507218104634077762312831185, −2.94038964370464569084863133877, −1.51867271711630592964624695034,
1.51867271711630592964624695034, 2.94038964370464569084863133877, 3.84507218104634077762312831185, 5.22921359046827082919555720214, 5.96544996271596631177230735419, 6.63652080618383635307639760474, 7.79780882030356989594681079953, 8.921140041973633294089213449529, 9.666111955655835904257090591884, 10.82492279477294017339874254148
