L(s) = 1 | + (−2.12 + 1.22i)2-s + (2.01 − 3.49i)4-s + (0.746 + 2.10i)5-s + 4.50i·7-s + 4.99i·8-s + (−4.17 − 3.56i)10-s − 2.19·11-s + (−3.25 − 1.87i)13-s + (−5.53 − 9.58i)14-s + (−2.09 − 3.63i)16-s + (−0.576 + 0.332i)17-s + (−3.79 − 2.13i)19-s + (8.86 + 1.64i)20-s + (4.67 − 2.70i)22-s + (0.422 + 0.244i)23-s + ⋯ |
L(s) = 1 | + (−1.50 + 0.868i)2-s + (1.00 − 1.74i)4-s + (0.333 + 0.942i)5-s + 1.70i·7-s + 1.76i·8-s + (−1.32 − 1.12i)10-s − 0.662·11-s + (−0.902 − 0.521i)13-s + (−1.47 − 2.56i)14-s + (−0.524 − 0.909i)16-s + (−0.139 + 0.0807i)17-s + (−0.871 − 0.490i)19-s + (1.98 + 0.367i)20-s + (0.997 − 0.575i)22-s + (0.0881 + 0.0508i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.375 + 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.375 + 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.139060 - 0.206447i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.139060 - 0.206447i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.746 - 2.10i)T \) |
| 19 | \( 1 + (3.79 + 2.13i)T \) |
good | 2 | \( 1 + (2.12 - 1.22i)T + (1 - 1.73i)T^{2} \) |
| 7 | \( 1 - 4.50iT - 7T^{2} \) |
| 11 | \( 1 + 2.19T + 11T^{2} \) |
| 13 | \( 1 + (3.25 + 1.87i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (0.576 - 0.332i)T + (8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-0.422 - 0.244i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.79 - 3.11i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 6.83T + 31T^{2} \) |
| 37 | \( 1 + 3.01iT - 37T^{2} \) |
| 41 | \( 1 + (-0.0362 - 0.0627i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.364 - 0.210i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.34 - 2.51i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2.26 + 1.30i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (6.26 + 10.8i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.53 - 6.11i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.95 + 2.86i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.48 - 6.03i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.56 + 1.47i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.66 - 9.81i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 15.6iT - 83T^{2} \) |
| 89 | \( 1 + (-0.668 + 1.15i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.79 + 2.19i)T + (48.5 - 84.0i)T^{2} \) |
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
−10.47898314590432113027570695855, −9.688942858277847696836505794042, −9.049136573058038987679244328048, −8.247875613297215586504960247955, −7.50955739512887592698506495996, −6.57177660677386949184600302580, −5.91500729533857915712176119535, −5.08768060994366530655604275781, −2.82982767945424879021353505550, −2.08993611049946386045330863730,
0.19873872659457771924841823517, 1.34968428045669544196767981155, 2.49033407054264998962547689802, 3.97257546041202941758264059825, 4.84577514129629569936521502934, 6.47111987557829730581965311049, 7.53373227854879957942158393203, 8.004514130602971296448634899139, 8.935577811620083141454771810075, 9.780301885353123415262606078542