| L(s) = 1 | − 0.625·2-s + 2.53·3-s − 1.60·4-s + 2.17·5-s − 1.58·6-s + 2.25·8-s + 3.41·9-s − 1.36·10-s + 2.29·11-s − 4.07·12-s − 3.85·13-s + 5.51·15-s + 1.80·16-s + 17-s − 2.13·18-s + 5.06·19-s − 3.50·20-s − 1.43·22-s + 5.90·23-s + 5.71·24-s − 0.261·25-s + 2.41·26-s + 1.05·27-s − 1.24·29-s − 3.44·30-s − 3.42·31-s − 5.64·32-s + ⋯ |
| L(s) = 1 | − 0.442·2-s + 1.46·3-s − 0.804·4-s + 0.973·5-s − 0.646·6-s + 0.798·8-s + 1.13·9-s − 0.430·10-s + 0.692·11-s − 1.17·12-s − 1.06·13-s + 1.42·15-s + 0.451·16-s + 0.242·17-s − 0.503·18-s + 1.16·19-s − 0.783·20-s − 0.306·22-s + 1.23·23-s + 1.16·24-s − 0.0522·25-s + 0.473·26-s + 0.203·27-s − 0.231·29-s − 0.629·30-s − 0.614·31-s − 0.997·32-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
| \(L(1)\) |
\(\approx\) |
\(2.066634929\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.066634929\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.625T + 2T^{2} \) |
| 3 | \( 1 - 2.53T + 3T^{2} \) |
| 5 | \( 1 - 2.17T + 5T^{2} \) |
| 11 | \( 1 - 2.29T + 11T^{2} \) |
| 13 | \( 1 + 3.85T + 13T^{2} \) |
| 19 | \( 1 - 5.06T + 19T^{2} \) |
| 23 | \( 1 - 5.90T + 23T^{2} \) |
| 29 | \( 1 + 1.24T + 29T^{2} \) |
| 31 | \( 1 + 3.42T + 31T^{2} \) |
| 37 | \( 1 - 3.99T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 + 11.0T + 43T^{2} \) |
| 47 | \( 1 - 7.10T + 47T^{2} \) |
| 53 | \( 1 - 13.2T + 53T^{2} \) |
| 59 | \( 1 + 9.38T + 59T^{2} \) |
| 61 | \( 1 + 10.6T + 61T^{2} \) |
| 67 | \( 1 + 13.3T + 67T^{2} \) |
| 71 | \( 1 - 1.40T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 + 0.561T + 79T^{2} \) |
| 83 | \( 1 + 0.620T + 83T^{2} \) |
| 89 | \( 1 - 4.34T + 89T^{2} \) |
| 97 | \( 1 + 5.92T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.669564986321087765389091757311, −9.369820910529457070895823539441, −8.849606668377617175313103314016, −7.75631302713661008174409313057, −7.22233605208952497997475149649, −5.73620545047872352976538153526, −4.74853299192523082925017830905, −3.61941164916407200341692116654, −2.56912404778294490590778620796, −1.36264375199200006387305686749,
1.36264375199200006387305686749, 2.56912404778294490590778620796, 3.61941164916407200341692116654, 4.74853299192523082925017830905, 5.73620545047872352976538153526, 7.22233605208952497997475149649, 7.75631302713661008174409313057, 8.849606668377617175313103314016, 9.369820910529457070895823539441, 9.669564986321087765389091757311
