| L(s) = 1 | + 1.19·2-s + 0.0386·3-s − 0.577·4-s + 5-s + 0.0461·6-s − 3.07·8-s − 2.99·9-s + 1.19·10-s + 4.80·11-s − 0.0223·12-s − 5.02·13-s + 0.0386·15-s − 2.51·16-s − 7.06·17-s − 3.57·18-s + 6.36·19-s − 0.577·20-s + 5.73·22-s + 5.82·23-s − 0.118·24-s + 25-s − 5.99·26-s − 0.232·27-s + 2.10·29-s + 0.0461·30-s + 4.56·31-s + 3.15·32-s + ⋯ |
| L(s) = 1 | + 0.843·2-s + 0.0223·3-s − 0.288·4-s + 0.447·5-s + 0.0188·6-s − 1.08·8-s − 0.999·9-s + 0.377·10-s + 1.44·11-s − 0.00644·12-s − 1.39·13-s + 0.00999·15-s − 0.628·16-s − 1.71·17-s − 0.843·18-s + 1.46·19-s − 0.129·20-s + 1.22·22-s + 1.21·23-s − 0.0242·24-s + 0.200·25-s − 1.17·26-s − 0.0446·27-s + 0.390·29-s + 0.00842·30-s + 0.820·31-s + 0.557·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9065 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9065 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.282286439\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.282286439\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 2 | \( 1 - 1.19T + 2T^{2} \) |
| 3 | \( 1 - 0.0386T + 3T^{2} \) |
| 11 | \( 1 - 4.80T + 11T^{2} \) |
| 13 | \( 1 + 5.02T + 13T^{2} \) |
| 17 | \( 1 + 7.06T + 17T^{2} \) |
| 19 | \( 1 - 6.36T + 19T^{2} \) |
| 23 | \( 1 - 5.82T + 23T^{2} \) |
| 29 | \( 1 - 2.10T + 29T^{2} \) |
| 31 | \( 1 - 4.56T + 31T^{2} \) |
| 41 | \( 1 + 9.81T + 41T^{2} \) |
| 43 | \( 1 - 1.88T + 43T^{2} \) |
| 47 | \( 1 + 11.2T + 47T^{2} \) |
| 53 | \( 1 - 1.66T + 53T^{2} \) |
| 59 | \( 1 + 2.79T + 59T^{2} \) |
| 61 | \( 1 + 2.43T + 61T^{2} \) |
| 67 | \( 1 + 8.99T + 67T^{2} \) |
| 71 | \( 1 + 1.51T + 71T^{2} \) |
| 73 | \( 1 - 2.42T + 73T^{2} \) |
| 79 | \( 1 - 7.52T + 79T^{2} \) |
| 83 | \( 1 + 0.303T + 83T^{2} \) |
| 89 | \( 1 - 16.1T + 89T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
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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
−7.63340788880951819258876653250, −6.61370665491315142820995658935, −6.48645634755021992760387179116, −5.48092055995251825284710404722, −4.86635160056178930810686356427, −4.49258458643034140944045839015, −3.34243186214924424344959293192, −2.93405615646695865339834014661, −1.95130120951083666528438266516, −0.62529498861111031488872470588,
0.62529498861111031488872470588, 1.95130120951083666528438266516, 2.93405615646695865339834014661, 3.34243186214924424344959293192, 4.49258458643034140944045839015, 4.86635160056178930810686356427, 5.48092055995251825284710404722, 6.48645634755021992760387179116, 6.61370665491315142820995658935, 7.63340788880951819258876653250
