| L(s) = 1 | + 2.02·2-s + 1.90·3-s + 2.08·4-s − 5-s + 3.84·6-s + 3.09·7-s + 0.175·8-s + 0.612·9-s − 2.02·10-s + 0.963·11-s + 3.96·12-s + 2.95·13-s + 6.25·14-s − 1.90·15-s − 3.81·16-s + 0.0389·17-s + 1.23·18-s − 2.08·20-s + 5.87·21-s + 1.94·22-s + 3.88·23-s + 0.334·24-s + 25-s + 5.98·26-s − 4.53·27-s + 6.45·28-s + 9.32·29-s + ⋯ |
| L(s) = 1 | + 1.42·2-s + 1.09·3-s + 1.04·4-s − 0.447·5-s + 1.56·6-s + 1.16·7-s + 0.0621·8-s + 0.204·9-s − 0.639·10-s + 0.290·11-s + 1.14·12-s + 0.820·13-s + 1.67·14-s − 0.490·15-s − 0.954·16-s + 0.00944·17-s + 0.291·18-s − 0.466·20-s + 1.28·21-s + 0.415·22-s + 0.809·23-s + 0.0681·24-s + 0.200·25-s + 1.17·26-s − 0.873·27-s + 1.21·28-s + 1.73·29-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
| \(L(1)\) |
\(\approx\) |
\(5.545275759\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.545275759\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.02T + 2T^{2} \) |
| 3 | \( 1 - 1.90T + 3T^{2} \) |
| 7 | \( 1 - 3.09T + 7T^{2} \) |
| 11 | \( 1 - 0.963T + 11T^{2} \) |
| 13 | \( 1 - 2.95T + 13T^{2} \) |
| 17 | \( 1 - 0.0389T + 17T^{2} \) |
| 23 | \( 1 - 3.88T + 23T^{2} \) |
| 29 | \( 1 - 9.32T + 29T^{2} \) |
| 31 | \( 1 - 9.37T + 31T^{2} \) |
| 37 | \( 1 - 1.11T + 37T^{2} \) |
| 41 | \( 1 + 12.0T + 41T^{2} \) |
| 43 | \( 1 + 4.95T + 43T^{2} \) |
| 47 | \( 1 - 8.01T + 47T^{2} \) |
| 53 | \( 1 + 6.05T + 53T^{2} \) |
| 59 | \( 1 + 2.77T + 59T^{2} \) |
| 61 | \( 1 + 10.0T + 61T^{2} \) |
| 67 | \( 1 + 9.34T + 67T^{2} \) |
| 71 | \( 1 + 12.5T + 71T^{2} \) |
| 73 | \( 1 - 0.897T + 73T^{2} \) |
| 79 | \( 1 + 8.89T + 79T^{2} \) |
| 83 | \( 1 + 9.60T + 83T^{2} \) |
| 89 | \( 1 - 8.74T + 89T^{2} \) |
| 97 | \( 1 + 7.78T + 97T^{2} \) |
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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
−8.810878788047566261909024605104, −8.571888192599617701763608763477, −7.75715580310272946711976118924, −6.75950371108429733331626508847, −5.92653215820498832295660685735, −4.79602423202589324781638631278, −4.40464154251027836341198366649, −3.32690586093229870034970102676, −2.80112984372276701231890967178, −1.49902817316852925318894985135,
1.49902817316852925318894985135, 2.80112984372276701231890967178, 3.32690586093229870034970102676, 4.40464154251027836341198366649, 4.79602423202589324781638631278, 5.92653215820498832295660685735, 6.75950371108429733331626508847, 7.75715580310272946711976118924, 8.571888192599617701763608763477, 8.810878788047566261909024605104
