| L(s) = 1 | + 1.53·2-s − 1.70·3-s + 0.369·4-s + 5-s − 2.63·6-s − 2.51·8-s − 0.0783·9-s + 1.53·10-s + 0.290·11-s − 0.630·12-s + 0.921·13-s − 1.70·15-s − 4.60·16-s − 4.97·17-s − 0.120·18-s + 6.04·19-s + 0.369·20-s + 0.447·22-s + 2.29·23-s + 4.29·24-s + 25-s + 1.41·26-s + 5.26·27-s + 29-s − 2.63·30-s − 10.0·31-s − 2.06·32-s + ⋯ |
| L(s) = 1 | + 1.08·2-s − 0.986·3-s + 0.184·4-s + 0.447·5-s − 1.07·6-s − 0.887·8-s − 0.0261·9-s + 0.486·10-s + 0.0876·11-s − 0.182·12-s + 0.255·13-s − 0.441·15-s − 1.15·16-s − 1.20·17-s − 0.0284·18-s + 1.38·19-s + 0.0825·20-s + 0.0954·22-s + 0.477·23-s + 0.875·24-s + 0.200·25-s + 0.278·26-s + 1.01·27-s + 0.185·29-s − 0.480·30-s − 1.80·31-s − 0.364·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.781767023\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.781767023\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 1.53T + 2T^{2} \) |
| 3 | \( 1 + 1.70T + 3T^{2} \) |
| 11 | \( 1 - 0.290T + 11T^{2} \) |
| 13 | \( 1 - 0.921T + 13T^{2} \) |
| 17 | \( 1 + 4.97T + 17T^{2} \) |
| 19 | \( 1 - 6.04T + 19T^{2} \) |
| 23 | \( 1 - 2.29T + 23T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 - 1.55T + 37T^{2} \) |
| 41 | \( 1 + 0.340T + 41T^{2} \) |
| 43 | \( 1 + 5.70T + 43T^{2} \) |
| 47 | \( 1 - 1.12T + 47T^{2} \) |
| 53 | \( 1 + 0.340T + 53T^{2} \) |
| 59 | \( 1 + 9.75T + 59T^{2} \) |
| 61 | \( 1 + 3.07T + 61T^{2} \) |
| 67 | \( 1 + 5.70T + 67T^{2} \) |
| 71 | \( 1 - 9.07T + 71T^{2} \) |
| 73 | \( 1 - 6.94T + 73T^{2} \) |
| 79 | \( 1 - 12.3T + 79T^{2} \) |
| 83 | \( 1 + 2.78T + 83T^{2} \) |
| 89 | \( 1 + 4.73T + 89T^{2} \) |
| 97 | \( 1 - 15.8T + 97T^{2} \) |
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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
−7.71983554201898612762952923797, −6.84442233808994624872928968177, −6.28323307723683848375078905184, −5.71225388993724954396940018456, −5.07964149422724064593600065280, −4.67904783222469732572063344348, −3.64090498196216460981247967540, −2.99789715820288830093119916784, −1.91796295955425399063877915868, −0.59100725320952630174146608315,
0.59100725320952630174146608315, 1.91796295955425399063877915868, 2.99789715820288830093119916784, 3.64090498196216460981247967540, 4.67904783222469732572063344348, 5.07964149422724064593600065280, 5.71225388993724954396940018456, 6.28323307723683848375078905184, 6.84442233808994624872928968177, 7.71983554201898612762952923797
