| L(s) = 1 | − 2.66·2-s + 2.65·3-s + 5.10·4-s − 0.476·5-s − 7.07·6-s − 1.66·7-s − 8.26·8-s + 4.05·9-s + 1.27·10-s − 0.629·11-s + 13.5·12-s − 2.17·13-s + 4.43·14-s − 1.26·15-s + 11.8·16-s + 1.89·17-s − 10.7·18-s + 4.87·19-s − 2.43·20-s − 4.41·21-s + 1.67·22-s − 2.42·23-s − 21.9·24-s − 4.77·25-s + 5.78·26-s + 2.79·27-s − 8.48·28-s + ⋯ |
| L(s) = 1 | − 1.88·2-s + 1.53·3-s + 2.55·4-s − 0.213·5-s − 2.88·6-s − 0.628·7-s − 2.92·8-s + 1.35·9-s + 0.401·10-s − 0.189·11-s + 3.91·12-s − 0.602·13-s + 1.18·14-s − 0.326·15-s + 2.95·16-s + 0.459·17-s − 2.54·18-s + 1.11·19-s − 0.544·20-s − 0.964·21-s + 0.357·22-s − 0.505·23-s − 4.48·24-s − 0.954·25-s + 1.13·26-s + 0.536·27-s − 1.60·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.156987958\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.156987958\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 6037 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.66T + 2T^{2} \) |
| 3 | \( 1 - 2.65T + 3T^{2} \) |
| 5 | \( 1 + 0.476T + 5T^{2} \) |
| 7 | \( 1 + 1.66T + 7T^{2} \) |
| 11 | \( 1 + 0.629T + 11T^{2} \) |
| 13 | \( 1 + 2.17T + 13T^{2} \) |
| 17 | \( 1 - 1.89T + 17T^{2} \) |
| 19 | \( 1 - 4.87T + 19T^{2} \) |
| 23 | \( 1 + 2.42T + 23T^{2} \) |
| 29 | \( 1 - 2.79T + 29T^{2} \) |
| 31 | \( 1 + 9.28T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 + 3.90T + 43T^{2} \) |
| 47 | \( 1 + 2.47T + 47T^{2} \) |
| 53 | \( 1 - 5.57T + 53T^{2} \) |
| 59 | \( 1 - 6.84T + 59T^{2} \) |
| 61 | \( 1 - 3.24T + 61T^{2} \) |
| 67 | \( 1 - 2.16T + 67T^{2} \) |
| 71 | \( 1 - 8.34T + 71T^{2} \) |
| 73 | \( 1 - 8.10T + 73T^{2} \) |
| 79 | \( 1 + 15.2T + 79T^{2} \) |
| 83 | \( 1 - 9.31T + 83T^{2} \) |
| 89 | \( 1 - 1.68T + 89T^{2} \) |
| 97 | \( 1 - 8.74T + 97T^{2} \) |
| 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
−8.050539734534456659041715017917, −7.66797870097022250263093841264, −7.26346473187448714586406762089, −6.36572871364762405700679937005, −5.47822308179446461746730087106, −3.96364572643483866476747991280, −3.20453330310962473502571322373, −2.53922539048706657423571407984, −1.85632933031145057663863663464, −0.67648650183751311800946252967,
0.67648650183751311800946252967, 1.85632933031145057663863663464, 2.53922539048706657423571407984, 3.20453330310962473502571322373, 3.96364572643483866476747991280, 5.47822308179446461746730087106, 6.36572871364762405700679937005, 7.26346473187448714586406762089, 7.66797870097022250263093841264, 8.050539734534456659041715017917
