| L(s) = 1 | − 2.60·2-s + 0.915·3-s + 4.76·4-s − 4.25·5-s − 2.38·6-s + 7-s − 7.20·8-s − 2.16·9-s + 11.0·10-s + 0.535·11-s + 4.36·12-s + 3.11·13-s − 2.60·14-s − 3.89·15-s + 9.19·16-s + 2.52·17-s + 5.62·18-s + 1.79·19-s − 20.2·20-s + 0.915·21-s − 1.39·22-s + 1.05·23-s − 6.59·24-s + 13.1·25-s − 8.11·26-s − 4.72·27-s + 4.76·28-s + ⋯ |
| L(s) = 1 | − 1.83·2-s + 0.528·3-s + 2.38·4-s − 1.90·5-s − 0.972·6-s + 0.377·7-s − 2.54·8-s − 0.720·9-s + 3.50·10-s + 0.161·11-s + 1.26·12-s + 0.864·13-s − 0.695·14-s − 1.00·15-s + 2.29·16-s + 0.611·17-s + 1.32·18-s + 0.411·19-s − 4.53·20-s + 0.199·21-s − 0.297·22-s + 0.219·23-s − 1.34·24-s + 2.62·25-s − 1.59·26-s − 0.909·27-s + 0.901·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5778022015\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5778022015\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 863 | \( 1 + T \) |
| good | 2 | \( 1 + 2.60T + 2T^{2} \) |
| 3 | \( 1 - 0.915T + 3T^{2} \) |
| 5 | \( 1 + 4.25T + 5T^{2} \) |
| 11 | \( 1 - 0.535T + 11T^{2} \) |
| 13 | \( 1 - 3.11T + 13T^{2} \) |
| 17 | \( 1 - 2.52T + 17T^{2} \) |
| 19 | \( 1 - 1.79T + 19T^{2} \) |
| 23 | \( 1 - 1.05T + 23T^{2} \) |
| 29 | \( 1 - 8.42T + 29T^{2} \) |
| 31 | \( 1 + 7.53T + 31T^{2} \) |
| 37 | \( 1 - 7.37T + 37T^{2} \) |
| 41 | \( 1 + 11.4T + 41T^{2} \) |
| 43 | \( 1 - 7.82T + 43T^{2} \) |
| 47 | \( 1 + 9.30T + 47T^{2} \) |
| 53 | \( 1 - 5.88T + 53T^{2} \) |
| 59 | \( 1 - 9.36T + 59T^{2} \) |
| 61 | \( 1 + 4.82T + 61T^{2} \) |
| 67 | \( 1 - 5.73T + 67T^{2} \) |
| 71 | \( 1 + 14.7T + 71T^{2} \) |
| 73 | \( 1 + 3.90T + 73T^{2} \) |
| 79 | \( 1 - 8.77T + 79T^{2} \) |
| 83 | \( 1 + 1.52T + 83T^{2} \) |
| 89 | \( 1 + 2.42T + 89T^{2} \) |
| 97 | \( 1 + 16.3T + 97T^{2} \) |
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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
−8.277861020576960237495547454788, −7.69277224559612570868162021812, −7.14699546515511882265651940400, −6.39528572221730966326424556588, −5.32885426773362908397718809653, −4.12982085386409906923242805101, −3.32168531919697802770109088192, −2.74874300807688689733611860770, −1.42369220380611437943570807493, −0.54951212864516851119464419818,
0.54951212864516851119464419818, 1.42369220380611437943570807493, 2.74874300807688689733611860770, 3.32168531919697802770109088192, 4.12982085386409906923242805101, 5.32885426773362908397718809653, 6.39528572221730966326424556588, 7.14699546515511882265651940400, 7.69277224559612570868162021812, 8.277861020576960237495547454788
