| L(s) = 1 | − 1.99·2-s − 0.822·3-s + 1.97·4-s − 2.77·5-s + 1.64·6-s + 2.78·7-s + 0.0449·8-s − 2.32·9-s + 5.53·10-s − 11-s − 1.62·12-s − 1.85·13-s − 5.55·14-s + 2.28·15-s − 4.04·16-s + 17-s + 4.63·18-s + 7.76·19-s − 5.48·20-s − 2.29·21-s + 1.99·22-s + 3.66·23-s − 0.0369·24-s + 2.69·25-s + 3.69·26-s + 4.37·27-s + 5.50·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.474·3-s + 0.988·4-s − 1.24·5-s + 0.669·6-s + 1.05·7-s + 0.0158·8-s − 0.774·9-s + 1.75·10-s − 0.301·11-s − 0.469·12-s − 0.513·13-s − 1.48·14-s + 0.589·15-s − 1.01·16-s + 0.242·17-s + 1.09·18-s + 1.78·19-s − 1.22·20-s − 0.499·21-s + 0.425·22-s + 0.763·23-s − 0.00754·24-s + 0.539·25-s + 0.723·26-s + 0.842·27-s + 1.04·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5848878991\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5848878991\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 + 1.99T + 2T^{2} \) |
| 3 | \( 1 + 0.822T + 3T^{2} \) |
| 5 | \( 1 + 2.77T + 5T^{2} \) |
| 7 | \( 1 - 2.78T + 7T^{2} \) |
| 13 | \( 1 + 1.85T + 13T^{2} \) |
| 19 | \( 1 - 7.76T + 19T^{2} \) |
| 23 | \( 1 - 3.66T + 23T^{2} \) |
| 29 | \( 1 - 5.63T + 29T^{2} \) |
| 31 | \( 1 - 8.12T + 31T^{2} \) |
| 37 | \( 1 + 11.9T + 37T^{2} \) |
| 41 | \( 1 - 6.89T + 41T^{2} \) |
| 47 | \( 1 - 8.06T + 47T^{2} \) |
| 53 | \( 1 + 0.364T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 + 1.10T + 61T^{2} \) |
| 67 | \( 1 + 12.2T + 67T^{2} \) |
| 71 | \( 1 - 4.56T + 71T^{2} \) |
| 73 | \( 1 - 1.34T + 73T^{2} \) |
| 79 | \( 1 + 10.0T + 79T^{2} \) |
| 83 | \( 1 + 1.45T + 83T^{2} \) |
| 89 | \( 1 + 7.60T + 89T^{2} \) |
| 97 | \( 1 - 8.66T + 97T^{2} \) |
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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
−7.912220351626620112506188373536, −7.38897852189311238877120479935, −6.92523560282537040083888305066, −5.73856585660430906078255770820, −4.96300023361732058580045583582, −4.52202655781031677522430057089, −3.31557552739235290077942262772, −2.52529990408682893501530672734, −1.22773663524541936566046258470, −0.56444326435329976561032657632,
0.56444326435329976561032657632, 1.22773663524541936566046258470, 2.52529990408682893501530672734, 3.31557552739235290077942262772, 4.52202655781031677522430057089, 4.96300023361732058580045583582, 5.73856585660430906078255770820, 6.92523560282537040083888305066, 7.38897852189311238877120479935, 7.912220351626620112506188373536
