| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 2·7-s − 8-s + 9-s + 11-s + 12-s + 4·13-s + 2·14-s + 16-s − 2·17-s − 18-s + 8·19-s − 2·21-s − 22-s + 4·23-s − 24-s − 5·25-s − 4·26-s + 27-s − 2·28-s − 6·31-s − 32-s + 33-s + 2·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.301·11-s + 0.288·12-s + 1.10·13-s + 0.534·14-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 1.83·19-s − 0.436·21-s − 0.213·22-s + 0.834·23-s − 0.204·24-s − 25-s − 0.784·26-s + 0.192·27-s − 0.377·28-s − 1.07·31-s − 0.176·32-s + 0.174·33-s + 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 1051 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.52804866719739, −13.75893941637630, −13.47692842189755, −12.91153398402562, −12.53387798327858, −11.61703020484402, −11.39014991140596, −10.94442224173963, −10.06316955664384, −9.819667349379895, −9.226360272024562, −8.950933508866282, −8.359219154604272, −7.648877887161934, −7.370040558488699, −6.734688370360716, −6.115740992275500, −5.715888008017529, −4.921518800004717, −4.052794801092758, −3.551609105584828, −3.035261866598587, −2.433465793149730, −1.485778896691085, −1.054267072692901, 0,
1.054267072692901, 1.485778896691085, 2.433465793149730, 3.035261866598587, 3.551609105584828, 4.052794801092758, 4.921518800004717, 5.715888008017529, 6.115740992275500, 6.734688370360716, 7.370040558488699, 7.648877887161934, 8.359219154604272, 8.950933508866282, 9.226360272024562, 9.819667349379895, 10.06316955664384, 10.94442224173963, 11.39014991140596, 11.61703020484402, 12.53387798327858, 12.91153398402562, 13.47692842189755, 13.75893941637630, 14.52804866719739
