| L(s) = 1 | − 2.01·2-s − 1.69·3-s + 2.07·4-s + 5-s + 3.42·6-s + 1.43·7-s − 0.153·8-s − 0.123·9-s − 2.01·10-s + 3.84·11-s − 3.52·12-s + 1.44·13-s − 2.89·14-s − 1.69·15-s − 3.84·16-s + 4.92·17-s + 0.248·18-s + 2.07·20-s − 2.43·21-s − 7.76·22-s + 5.85·23-s + 0.260·24-s + 25-s − 2.92·26-s + 5.29·27-s + 2.97·28-s + 7.03·29-s + ⋯ |
| L(s) = 1 | − 1.42·2-s − 0.979·3-s + 1.03·4-s + 0.447·5-s + 1.39·6-s + 0.541·7-s − 0.0543·8-s − 0.0411·9-s − 0.638·10-s + 1.15·11-s − 1.01·12-s + 0.401·13-s − 0.773·14-s − 0.437·15-s − 0.960·16-s + 1.19·17-s + 0.0586·18-s + 0.464·20-s − 0.530·21-s − 1.65·22-s + 1.22·23-s + 0.0532·24-s + 0.200·25-s − 0.572·26-s + 1.01·27-s + 0.562·28-s + 1.30·29-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
| \(L(1)\) |
\(\approx\) |
\(0.7701896933\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7701896933\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.01T + 2T^{2} \) |
| 3 | \( 1 + 1.69T + 3T^{2} \) |
| 7 | \( 1 - 1.43T + 7T^{2} \) |
| 11 | \( 1 - 3.84T + 11T^{2} \) |
| 13 | \( 1 - 1.44T + 13T^{2} \) |
| 17 | \( 1 - 4.92T + 17T^{2} \) |
| 23 | \( 1 - 5.85T + 23T^{2} \) |
| 29 | \( 1 - 7.03T + 29T^{2} \) |
| 31 | \( 1 + 1.72T + 31T^{2} \) |
| 37 | \( 1 - 11.6T + 37T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 43 | \( 1 + 1.99T + 43T^{2} \) |
| 47 | \( 1 + 2.91T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 + 4.52T + 59T^{2} \) |
| 61 | \( 1 - 0.465T + 61T^{2} \) |
| 67 | \( 1 - 2.72T + 67T^{2} \) |
| 71 | \( 1 - 6.03T + 71T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 + 15.7T + 83T^{2} \) |
| 89 | \( 1 + 6.11T + 89T^{2} \) |
| 97 | \( 1 - 4.35T + 97T^{2} \) |
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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
−9.403071396929432161820531811263, −8.482726474719399653290193210404, −7.982039502268113486048528821841, −6.79806317350676531925198029557, −6.39885819300729898352695737365, −5.33318930890897247032327041429, −4.56880856338782629443840299302, −3.09691532116877329877284534606, −1.55765458724216661694713482607, −0.873049828857164976876205312634,
0.873049828857164976876205312634, 1.55765458724216661694713482607, 3.09691532116877329877284534606, 4.56880856338782629443840299302, 5.33318930890897247032327041429, 6.39885819300729898352695737365, 6.79806317350676531925198029557, 7.982039502268113486048528821841, 8.482726474719399653290193210404, 9.403071396929432161820531811263
