| L(s) = 1 | + 0.971·2-s − 1.65·3-s − 1.05·4-s + 2.99·5-s − 1.60·6-s − 2.96·8-s − 0.275·9-s + 2.90·10-s + 1.75·11-s + 1.74·12-s + 1.15·13-s − 4.93·15-s − 0.775·16-s + 17-s − 0.268·18-s + 8.19·19-s − 3.15·20-s + 1.70·22-s + 2.00·23-s + 4.90·24-s + 3.95·25-s + 1.12·26-s + 5.40·27-s − 8.40·29-s − 4.80·30-s + 1.49·31-s + 5.18·32-s + ⋯ |
| L(s) = 1 | + 0.687·2-s − 0.952·3-s − 0.527·4-s + 1.33·5-s − 0.654·6-s − 1.04·8-s − 0.0919·9-s + 0.919·10-s + 0.529·11-s + 0.502·12-s + 0.321·13-s − 1.27·15-s − 0.193·16-s + 0.242·17-s − 0.0632·18-s + 1.87·19-s − 0.706·20-s + 0.363·22-s + 0.417·23-s + 1.00·24-s + 0.791·25-s + 0.220·26-s + 1.04·27-s − 1.56·29-s − 0.876·30-s + 0.267·31-s + 0.916·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.680702241\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.680702241\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 - 0.971T + 2T^{2} \) |
| 3 | \( 1 + 1.65T + 3T^{2} \) |
| 5 | \( 1 - 2.99T + 5T^{2} \) |
| 11 | \( 1 - 1.75T + 11T^{2} \) |
| 13 | \( 1 - 1.15T + 13T^{2} \) |
| 19 | \( 1 - 8.19T + 19T^{2} \) |
| 23 | \( 1 - 2.00T + 23T^{2} \) |
| 29 | \( 1 + 8.40T + 29T^{2} \) |
| 31 | \( 1 - 1.49T + 31T^{2} \) |
| 37 | \( 1 - 0.0964T + 37T^{2} \) |
| 41 | \( 1 - 7.95T + 41T^{2} \) |
| 43 | \( 1 - 8.56T + 43T^{2} \) |
| 47 | \( 1 - 1.22T + 47T^{2} \) |
| 53 | \( 1 - 8.80T + 53T^{2} \) |
| 59 | \( 1 + 2.31T + 59T^{2} \) |
| 61 | \( 1 - 8.19T + 61T^{2} \) |
| 67 | \( 1 + 0.640T + 67T^{2} \) |
| 71 | \( 1 - 11.5T + 71T^{2} \) |
| 73 | \( 1 + 16.5T + 73T^{2} \) |
| 79 | \( 1 + 8.79T + 79T^{2} \) |
| 83 | \( 1 + 12.0T + 83T^{2} \) |
| 89 | \( 1 + 0.421T + 89T^{2} \) |
| 97 | \( 1 - 4.53T + 97T^{2} \) |
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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
−10.15423573266768130370693336373, −9.419589310953153688882029578698, −8.863581473446365218983561435964, −7.39706026419115638026745034276, −6.24121786202597903324399891248, −5.61329052386683780439908196628, −5.26622766134505446822553286656, −3.99774488720279739736111009503, −2.78618949162375166644393459371, −1.06733659792229569683590372176,
1.06733659792229569683590372176, 2.78618949162375166644393459371, 3.99774488720279739736111009503, 5.26622766134505446822553286656, 5.61329052386683780439908196628, 6.24121786202597903324399891248, 7.39706026419115638026745034276, 8.863581473446365218983561435964, 9.419589310953153688882029578698, 10.15423573266768130370693336373
