| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 7-s + 8-s + 9-s + 10-s − 4·11-s + 12-s − 2·13-s + 14-s + 15-s + 16-s + 17-s + 18-s + 4·19-s + 20-s + 21-s − 4·22-s + 8·23-s + 24-s + 25-s − 2·26-s + 27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s + 0.288·12-s − 0.554·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.917·19-s + 0.223·20-s + 0.218·21-s − 0.852·22-s + 1.66·23-s + 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.854682430\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.854682430\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.68017142186039, −13.10464404638792, −12.85615738228401, −12.46683960089896, −11.63704726962211, −11.31662881774289, −10.73826672686368, −10.29469362412058, −9.639554702953989, −9.401891942077475, −8.620996410570280, −8.144196767658846, −7.541886544822074, −7.217238280127678, −6.742257566562262, −5.784566887400274, −5.506643723255510, −4.989996973633639, −4.505495832804927, −3.809448450756095, −3.075102800879225, −2.669893813751381, −2.243599223309768, −1.375520758646098, −0.7000837864831691,
0.7000837864831691, 1.375520758646098, 2.243599223309768, 2.669893813751381, 3.075102800879225, 3.809448450756095, 4.505495832804927, 4.989996973633639, 5.506643723255510, 5.784566887400274, 6.742257566562262, 7.217238280127678, 7.541886544822074, 8.144196767658846, 8.620996410570280, 9.401891942077475, 9.639554702953989, 10.29469362412058, 10.73826672686368, 11.31662881774289, 11.63704726962211, 12.46683960089896, 12.85615738228401, 13.10464404638792, 13.68017142186039
