| L(s) = 1 | − 2-s − 1.81·3-s + 4-s + 1.81·6-s − 2.52·7-s − 8-s + 0.289·9-s − 2.91·11-s − 1.81·12-s − 3.10·13-s + 2.52·14-s + 16-s − 6.91·17-s − 0.289·18-s + 1.81·19-s + 4.57·21-s + 2.91·22-s + 5.68·23-s + 1.81·24-s + 3.10·26-s + 4.91·27-s − 2.52·28-s + 29-s − 8.12·31-s − 32-s + 5.28·33-s + 6.91·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.04·3-s + 0.5·4-s + 0.740·6-s − 0.954·7-s − 0.353·8-s + 0.0963·9-s − 0.879·11-s − 0.523·12-s − 0.860·13-s + 0.674·14-s + 0.250·16-s − 1.67·17-s − 0.0681·18-s + 0.416·19-s + 0.999·21-s + 0.621·22-s + 1.18·23-s + 0.370·24-s + 0.608·26-s + 0.946·27-s − 0.477·28-s + 0.185·29-s − 1.45·31-s − 0.176·32-s + 0.920·33-s + 1.18·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3015612892\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3015612892\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + 1.81T + 3T^{2} \) |
| 7 | \( 1 + 2.52T + 7T^{2} \) |
| 11 | \( 1 + 2.91T + 11T^{2} \) |
| 13 | \( 1 + 3.10T + 13T^{2} \) |
| 17 | \( 1 + 6.91T + 17T^{2} \) |
| 19 | \( 1 - 1.81T + 19T^{2} \) |
| 23 | \( 1 - 5.68T + 23T^{2} \) |
| 31 | \( 1 + 8.12T + 31T^{2} \) |
| 37 | \( 1 + 7.39T + 37T^{2} \) |
| 41 | \( 1 - 8.39T + 41T^{2} \) |
| 43 | \( 1 - 5.04T + 43T^{2} \) |
| 47 | \( 1 + 4.86T + 47T^{2} \) |
| 53 | \( 1 + 5.30T + 53T^{2} \) |
| 59 | \( 1 - 2.60T + 59T^{2} \) |
| 61 | \( 1 + 10.0T + 61T^{2} \) |
| 67 | \( 1 - 2.68T + 67T^{2} \) |
| 71 | \( 1 - 5.68T + 71T^{2} \) |
| 73 | \( 1 - 15.0T + 73T^{2} \) |
| 79 | \( 1 + 1.73T + 79T^{2} \) |
| 83 | \( 1 - 13.1T + 83T^{2} \) |
| 89 | \( 1 - 5.23T + 89T^{2} \) |
| 97 | \( 1 + 1.27T + 97T^{2} \) |
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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
−9.421216691389310699335740059256, −9.007245624309739732261719946411, −7.82204478911381106852760462014, −6.97017807543440544064693025036, −6.43234758803930161092661471325, −5.46056182396846015839013399038, −4.75519920239897225798256834116, −3.23959507433057884282487036503, −2.25531568432971367181109922216, −0.42843226143094794011313825231,
0.42843226143094794011313825231, 2.25531568432971367181109922216, 3.23959507433057884282487036503, 4.75519920239897225798256834116, 5.46056182396846015839013399038, 6.43234758803930161092661471325, 6.97017807543440544064693025036, 7.82204478911381106852760462014, 9.007245624309739732261719946411, 9.421216691389310699335740059256
