| L(s) = 1 | + 2-s + 3-s + 4-s + 2.39·5-s + 6-s − 1.97·7-s + 8-s + 9-s + 2.39·10-s + 1.05·11-s + 12-s + 1.63·13-s − 1.97·14-s + 2.39·15-s + 16-s + 4.59·17-s + 18-s − 19-s + 2.39·20-s − 1.97·21-s + 1.05·22-s − 1.09·23-s + 24-s + 0.717·25-s + 1.63·26-s + 27-s − 1.97·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.06·5-s + 0.408·6-s − 0.747·7-s + 0.353·8-s + 0.333·9-s + 0.756·10-s + 0.318·11-s + 0.288·12-s + 0.453·13-s − 0.528·14-s + 0.617·15-s + 0.250·16-s + 1.11·17-s + 0.235·18-s − 0.229·19-s + 0.534·20-s − 0.431·21-s + 0.225·22-s − 0.227·23-s + 0.204·24-s + 0.143·25-s + 0.320·26-s + 0.192·27-s − 0.373·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.123066273\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.123066273\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 - T \) |
| good | 5 | \( 1 - 2.39T + 5T^{2} \) |
| 7 | \( 1 + 1.97T + 7T^{2} \) |
| 11 | \( 1 - 1.05T + 11T^{2} \) |
| 13 | \( 1 - 1.63T + 13T^{2} \) |
| 17 | \( 1 - 4.59T + 17T^{2} \) |
| 23 | \( 1 + 1.09T + 23T^{2} \) |
| 29 | \( 1 - 8.10T + 29T^{2} \) |
| 31 | \( 1 - 9.13T + 31T^{2} \) |
| 37 | \( 1 + 4.12T + 37T^{2} \) |
| 41 | \( 1 + 9.15T + 41T^{2} \) |
| 43 | \( 1 + 0.538T + 43T^{2} \) |
| 47 | \( 1 + 2.73T + 47T^{2} \) |
| 59 | \( 1 - 9.25T + 59T^{2} \) |
| 61 | \( 1 - 9.85T + 61T^{2} \) |
| 67 | \( 1 + 5.45T + 67T^{2} \) |
| 71 | \( 1 - 2.55T + 71T^{2} \) |
| 73 | \( 1 + 12.5T + 73T^{2} \) |
| 79 | \( 1 + 7.92T + 79T^{2} \) |
| 83 | \( 1 - 4.72T + 83T^{2} \) |
| 89 | \( 1 + 3.82T + 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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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
−8.212691594329148629790762463079, −7.11973294310082733037741991499, −6.50031528717214983182755570687, −6.02626763227423831639090003572, −5.23931383844003878459094023935, −4.41400289112177188939199448539, −3.46268337224839660270401603542, −2.95095243459849188908379557753, −2.03914971134598557868817380806, −1.11504197651370158066496049723,
1.11504197651370158066496049723, 2.03914971134598557868817380806, 2.95095243459849188908379557753, 3.46268337224839660270401603542, 4.41400289112177188939199448539, 5.23931383844003878459094023935, 6.02626763227423831639090003572, 6.50031528717214983182755570687, 7.11973294310082733037741991499, 8.212691594329148629790762463079
