L(s) = 1 | + (0.431 + 0.747i)2-s + (1.53 + 2.66i)3-s + (0.627 − 1.08i)4-s + (−1.32 + 2.30i)6-s − 0.566·7-s + 2.80·8-s + (−3.24 + 5.61i)9-s − 1.91·11-s + 3.86·12-s + (0.0972 − 0.168i)13-s + (−0.244 − 0.423i)14-s + (−0.0438 − 0.0760i)16-s + (2.64 + 4.58i)17-s − 5.59·18-s + (−2.36 + 3.65i)19-s + ⋯ |
L(s) = 1 | + (0.305 + 0.528i)2-s + (0.888 + 1.53i)3-s + (0.313 − 0.543i)4-s + (−0.542 + 0.939i)6-s − 0.214·7-s + 0.993·8-s + (−1.08 + 1.87i)9-s − 0.576·11-s + 1.11·12-s + (0.0269 − 0.0467i)13-s + (−0.0653 − 0.113i)14-s + (−0.0109 − 0.0190i)16-s + (0.642 + 1.11i)17-s − 1.31·18-s + (−0.543 + 0.839i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.257 - 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.257 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.40835 + 1.83186i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.40835 + 1.83186i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 + (2.36 - 3.65i)T \) |
good | 2 | \( 1 + (-0.431 - 0.747i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (-1.53 - 2.66i)T + (-1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + 0.566T + 7T^{2} \) |
| 11 | \( 1 + 1.91T + 11T^{2} \) |
| 13 | \( 1 + (-0.0972 + 0.168i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.64 - 4.58i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-1.68 + 2.92i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.36 + 7.56i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 5.65T + 31T^{2} \) |
| 37 | \( 1 + 0.955T + 37T^{2} \) |
| 41 | \( 1 + (5.02 + 8.70i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.46 + 4.27i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.41 + 7.65i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (4.10 - 7.10i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (1.85 + 3.21i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.75 + 3.04i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.02 + 3.50i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (2.59 + 4.49i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.30 - 7.45i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.31 + 5.73i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 4.51T + 83T^{2} \) |
| 89 | \( 1 + (1.68 - 2.92i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.59 - 13.1i)T + (-48.5 + 84.0i)T^{2} \) |
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
−10.65480735766685308064667140241, −10.39288113865843560408729062916, −9.700117514500822263709627148382, −8.483491013175672646245312676873, −7.905836640142327887627079744749, −6.43358482037000427754616725896, −5.46526190257482908782091616849, −4.55098067483899670421478898733, −3.61101857869628465214667276316, −2.27641312930943371682793972174,
1.35524937247830206127772435533, 2.73902262646111318230320050660, 3.13637581438554406600844331283, 4.85185989622309320707305252512, 6.51163171964537117209906037155, 7.16455025721099274216444148582, 7.937116863203389126630577483602, 8.659006373156005730385078587686, 9.810042619696132495963780606909, 11.12121180781366240667685374553