| L(s) = 1 | − 1.38·2-s − 3.31·3-s − 0.0703·4-s − 5-s + 4.60·6-s − 1.97·7-s + 2.87·8-s + 8.00·9-s + 1.38·10-s + 4.53·11-s + 0.233·12-s − 2.78·13-s + 2.74·14-s + 3.31·15-s − 3.85·16-s + 0.944·17-s − 11.1·18-s + 0.0703·20-s + 6.56·21-s − 6.29·22-s + 6.80·23-s − 9.54·24-s + 25-s + 3.87·26-s − 16.6·27-s + 0.139·28-s + 2.80·29-s + ⋯ |
| L(s) = 1 | − 0.982·2-s − 1.91·3-s − 0.0351·4-s − 0.447·5-s + 1.88·6-s − 0.748·7-s + 1.01·8-s + 2.66·9-s + 0.439·10-s + 1.36·11-s + 0.0673·12-s − 0.773·13-s + 0.734·14-s + 0.856·15-s − 0.963·16-s + 0.229·17-s − 2.62·18-s + 0.0157·20-s + 1.43·21-s − 1.34·22-s + 1.41·23-s − 1.94·24-s + 0.200·25-s + 0.759·26-s − 3.19·27-s + 0.0263·28-s + 0.520·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3065925434\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3065925434\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 1.38T + 2T^{2} \) |
| 3 | \( 1 + 3.31T + 3T^{2} \) |
| 7 | \( 1 + 1.97T + 7T^{2} \) |
| 11 | \( 1 - 4.53T + 11T^{2} \) |
| 13 | \( 1 + 2.78T + 13T^{2} \) |
| 17 | \( 1 - 0.944T + 17T^{2} \) |
| 23 | \( 1 - 6.80T + 23T^{2} \) |
| 29 | \( 1 - 2.80T + 29T^{2} \) |
| 31 | \( 1 + 1.55T + 31T^{2} \) |
| 37 | \( 1 + 1.67T + 37T^{2} \) |
| 41 | \( 1 + 4.34T + 41T^{2} \) |
| 43 | \( 1 - 0.230T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 + 1.87T + 53T^{2} \) |
| 59 | \( 1 - 13.4T + 59T^{2} \) |
| 61 | \( 1 + 8.56T + 61T^{2} \) |
| 67 | \( 1 - 3.50T + 67T^{2} \) |
| 71 | \( 1 - 1.14T + 71T^{2} \) |
| 73 | \( 1 + 8.60T + 73T^{2} \) |
| 79 | \( 1 - 3.70T + 79T^{2} \) |
| 83 | \( 1 - 13.1T + 83T^{2} \) |
| 89 | \( 1 + 11.7T + 89T^{2} \) |
| 97 | \( 1 - 4.65T + 97T^{2} \) |
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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
−9.557608868313235323965118932607, −8.654540376246876778560395338923, −7.45981528411911400958989535676, −6.87117051358083173186155804161, −6.34492786319166362282466877037, −5.14735240070394210519681183786, −4.59394483942419866555739683936, −3.57766416898384045398258269275, −1.48156027491688597429237378835, −0.52676908828243685798429048782,
0.52676908828243685798429048782, 1.48156027491688597429237378835, 3.57766416898384045398258269275, 4.59394483942419866555739683936, 5.14735240070394210519681183786, 6.34492786319166362282466877037, 6.87117051358083173186155804161, 7.45981528411911400958989535676, 8.654540376246876778560395338923, 9.557608868313235323965118932607
