| L(s) = 1 | + 2.58·2-s − 0.247·3-s + 4.67·4-s + 5-s − 0.639·6-s − 0.401·7-s + 6.90·8-s − 2.93·9-s + 2.58·10-s + 5.19·11-s − 1.15·12-s + 2.88·13-s − 1.03·14-s − 0.247·15-s + 8.48·16-s − 4.14·17-s − 7.59·18-s + 4.67·20-s + 0.0994·21-s + 13.4·22-s + 3.35·23-s − 1.71·24-s + 25-s + 7.46·26-s + 1.47·27-s − 1.87·28-s + 6.58·29-s + ⋯ |
| L(s) = 1 | + 1.82·2-s − 0.143·3-s + 2.33·4-s + 0.447·5-s − 0.261·6-s − 0.151·7-s + 2.44·8-s − 0.979·9-s + 0.816·10-s + 1.56·11-s − 0.334·12-s + 0.801·13-s − 0.277·14-s − 0.0639·15-s + 2.12·16-s − 1.00·17-s − 1.78·18-s + 1.04·20-s + 0.0217·21-s + 2.85·22-s + 0.700·23-s − 0.349·24-s + 0.200·25-s + 1.46·26-s + 0.283·27-s − 0.354·28-s + 1.22·29-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
| \(L(1)\) |
\(\approx\) |
\(5.658711757\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.658711757\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.58T + 2T^{2} \) |
| 3 | \( 1 + 0.247T + 3T^{2} \) |
| 7 | \( 1 + 0.401T + 7T^{2} \) |
| 11 | \( 1 - 5.19T + 11T^{2} \) |
| 13 | \( 1 - 2.88T + 13T^{2} \) |
| 17 | \( 1 + 4.14T + 17T^{2} \) |
| 23 | \( 1 - 3.35T + 23T^{2} \) |
| 29 | \( 1 - 6.58T + 29T^{2} \) |
| 31 | \( 1 + 6.26T + 31T^{2} \) |
| 37 | \( 1 - 1.14T + 37T^{2} \) |
| 41 | \( 1 - 2.85T + 41T^{2} \) |
| 43 | \( 1 + 12.4T + 43T^{2} \) |
| 47 | \( 1 - 6.76T + 47T^{2} \) |
| 53 | \( 1 - 12.3T + 53T^{2} \) |
| 59 | \( 1 + 1.51T + 59T^{2} \) |
| 61 | \( 1 + 6.98T + 61T^{2} \) |
| 67 | \( 1 + 0.757T + 67T^{2} \) |
| 71 | \( 1 + 8.33T + 71T^{2} \) |
| 73 | \( 1 + 9.91T + 73T^{2} \) |
| 79 | \( 1 - 3.18T + 79T^{2} \) |
| 83 | \( 1 - 5.51T + 83T^{2} \) |
| 89 | \( 1 + 12.5T + 89T^{2} \) |
| 97 | \( 1 + 8.89T + 97T^{2} \) |
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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
−9.126388725522001069956382405598, −8.589871121740432488260009989728, −7.14466321135617480008563634895, −6.45663730964261979945099155262, −6.05354913461931212475572112410, −5.18319712135113712937845911187, −4.31222401463205090976792453221, −3.51553595036025410872694577195, −2.66988021321079334180976351302, −1.49970302379766694408085521015,
1.49970302379766694408085521015, 2.66988021321079334180976351302, 3.51553595036025410872694577195, 4.31222401463205090976792453221, 5.18319712135113712937845911187, 6.05354913461931212475572112410, 6.45663730964261979945099155262, 7.14466321135617480008563634895, 8.589871121740432488260009989728, 9.126388725522001069956382405598
