L(s) = 1 | + 3·5-s + (0.5 − 2.59i)7-s − 5.19i·11-s + (−1.5 + 2.59i)17-s + (1.5 − 0.866i)19-s − 5.19i·23-s + 4·25-s + (1.5 − 0.866i)31-s + (1.5 − 7.79i)35-s + (−3.5 − 6.06i)37-s + (−3 + 5.19i)41-s + (−2 − 3.46i)43-s + (−1.5 + 2.59i)47-s + (−6.5 − 2.59i)49-s + (−4.5 − 2.59i)53-s + ⋯ |
L(s) = 1 | + 1.34·5-s + (0.188 − 0.981i)7-s − 1.56i·11-s + (−0.363 + 0.630i)17-s + (0.344 − 0.198i)19-s − 1.08i·23-s + 0.800·25-s + (0.269 − 0.155i)31-s + (0.253 − 1.31i)35-s + (−0.575 − 0.996i)37-s + (−0.468 + 0.811i)41-s + (−0.304 − 0.528i)43-s + (−0.218 + 0.378i)47-s + (−0.928 − 0.371i)49-s + (−0.618 − 0.356i)53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.110 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.110 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.162030126\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.162030126\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.5 + 2.59i)T \) |
good | 5 | \( 1 - 3T + 5T^{2} \) |
| 11 | \( 1 + 5.19iT - 11T^{2} \) |
| 13 | \( 1 + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.5 + 0.866i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 5.19iT - 23T^{2} \) |
| 29 | \( 1 + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.5 + 0.866i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.5 + 6.06i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (3 - 5.19i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2 + 3.46i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (1.5 - 2.59i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (4.5 + 2.59i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.5 - 2.59i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-10.5 - 6.06i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.5 + 4.33i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 10.3iT - 71T^{2} \) |
| 73 | \( 1 + (10.5 + 6.06i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-6 - 10.3i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-4.5 - 7.79i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-6 + 3.46i)T + (48.5 - 84.0i)T^{2} \) |
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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
−8.752041133443531888829114951903, −8.281961484380062495765253375445, −7.17672664451440366856367127570, −6.37316710686069521372408838946, −5.83090593779785982239475053500, −4.94583455079329553210834293689, −3.92082740073661750972298074271, −2.95778013428446797122959932101, −1.84229964395840046097251360452, −0.72023189104930282554618849696,
1.65958147445020219223360497828, 2.17600248229156634412609471073, 3.24130408263612626306112598333, 4.72618983539529281305414390028, 5.20773698191791038252651568163, 6.02066750183582009935559786758, 6.80562475031063690875275129489, 7.58524307345143014828026318324, 8.621309308469259514613133221275, 9.364789242020456460814057884160