| L(s) = 1 | − 2.57·2-s − 0.937·3-s + 4.64·4-s − 3.87·5-s + 2.41·6-s + 7-s − 6.81·8-s − 2.12·9-s + 9.99·10-s + 4.77·11-s − 4.35·12-s − 5.45·13-s − 2.57·14-s + 3.63·15-s + 8.28·16-s − 2.70·17-s + 5.46·18-s − 0.0387·19-s − 18.0·20-s − 0.937·21-s − 12.2·22-s + 8.78·23-s + 6.39·24-s + 10.0·25-s + 14.0·26-s + 4.80·27-s + 4.64·28-s + ⋯ |
| L(s) = 1 | − 1.82·2-s − 0.541·3-s + 2.32·4-s − 1.73·5-s + 0.986·6-s + 0.377·7-s − 2.41·8-s − 0.706·9-s + 3.16·10-s + 1.43·11-s − 1.25·12-s − 1.51·13-s − 0.688·14-s + 0.939·15-s + 2.07·16-s − 0.655·17-s + 1.28·18-s − 0.00888·19-s − 4.02·20-s − 0.204·21-s − 2.62·22-s + 1.83·23-s + 1.30·24-s + 2.00·25-s + 2.75·26-s + 0.924·27-s + 0.877·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2229073223\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2229073223\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 863 | \( 1 + T \) |
| good | 2 | \( 1 + 2.57T + 2T^{2} \) |
| 3 | \( 1 + 0.937T + 3T^{2} \) |
| 5 | \( 1 + 3.87T + 5T^{2} \) |
| 11 | \( 1 - 4.77T + 11T^{2} \) |
| 13 | \( 1 + 5.45T + 13T^{2} \) |
| 17 | \( 1 + 2.70T + 17T^{2} \) |
| 19 | \( 1 + 0.0387T + 19T^{2} \) |
| 23 | \( 1 - 8.78T + 23T^{2} \) |
| 29 | \( 1 + 5.81T + 29T^{2} \) |
| 31 | \( 1 - 7.97T + 31T^{2} \) |
| 37 | \( 1 + 10.0T + 37T^{2} \) |
| 41 | \( 1 + 1.45T + 41T^{2} \) |
| 43 | \( 1 - 6.85T + 43T^{2} \) |
| 47 | \( 1 + 7.22T + 47T^{2} \) |
| 53 | \( 1 + 8.65T + 53T^{2} \) |
| 59 | \( 1 + 9.38T + 59T^{2} \) |
| 61 | \( 1 - 13.2T + 61T^{2} \) |
| 67 | \( 1 + 0.538T + 67T^{2} \) |
| 71 | \( 1 - 4.14T + 71T^{2} \) |
| 73 | \( 1 - 0.785T + 73T^{2} \) |
| 79 | \( 1 - 6.77T + 79T^{2} \) |
| 83 | \( 1 + 10.7T + 83T^{2} \) |
| 89 | \( 1 + 12.2T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
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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
−8.106619257246634243869509281660, −7.53709166378521402953944293113, −6.83775018286369531533617230354, −6.58468039127344334449287032654, −5.19733046460698369249967701671, −4.49174786151603865513107255301, −3.40356993382563071214824126383, −2.57925924905287333987249543142, −1.31786360756810388334900661516, −0.36648567336228768851790083608,
0.36648567336228768851790083608, 1.31786360756810388334900661516, 2.57925924905287333987249543142, 3.40356993382563071214824126383, 4.49174786151603865513107255301, 5.19733046460698369249967701671, 6.58468039127344334449287032654, 6.83775018286369531533617230354, 7.53709166378521402953944293113, 8.106619257246634243869509281660
