L(s) = 1 | − 1.68·2-s + (−1.34 + 2.32i)3-s + 0.851·4-s + (2.27 − 3.93i)6-s + (2.52 − 4.37i)7-s + 1.93·8-s + (−2.11 − 3.66i)9-s + (−0.726 − 1.25i)11-s + (−1.14 + 1.98i)12-s + (−0.381 − 0.661i)13-s + (−4.26 + 7.38i)14-s − 4.97·16-s + (−2.68 + 4.65i)17-s + (3.57 + 6.18i)18-s + (0.756 − 1.31i)19-s + ⋯ |
L(s) = 1 | − 1.19·2-s + (−0.776 + 1.34i)3-s + 0.425·4-s + (0.926 − 1.60i)6-s + (0.954 − 1.65i)7-s + 0.685·8-s + (−0.704 − 1.22i)9-s + (−0.218 − 0.379i)11-s + (−0.330 + 0.572i)12-s + (−0.105 − 0.183i)13-s + (−1.14 + 1.97i)14-s − 1.24·16-s + (−0.651 + 1.12i)17-s + (0.841 + 1.45i)18-s + (0.173 − 0.300i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.914 + 0.404i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.914 + 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0100044 - 0.0473853i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0100044 - 0.0473853i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 31 | \( 1 + (-1.61 - 5.32i)T \) |
good | 2 | \( 1 + 1.68T + 2T^{2} \) |
| 3 | \( 1 + (1.34 - 2.32i)T + (-1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + (-2.52 + 4.37i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.726 + 1.25i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.381 + 0.661i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (2.68 - 4.65i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.756 + 1.31i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 3.39T + 23T^{2} \) |
| 29 | \( 1 + 5.39T + 29T^{2} \) |
| 37 | \( 1 + (4.95 - 8.58i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.604 + 1.04i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.92 + 5.07i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 5.15T + 47T^{2} \) |
| 53 | \( 1 + (-0.783 - 1.35i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (5.78 - 10.0i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + 11.7T + 61T^{2} \) |
| 67 | \( 1 + (2.64 + 4.58i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.09 - 10.5i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.08 + 3.61i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.868 - 1.50i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.88 - 8.46i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 11.9T + 89T^{2} \) |
| 97 | \( 1 - 2.12T + 97T^{2} \) |
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
−10.58885099428871470625377898120, −10.22984709602109853385278081648, −9.255868238510070607612647421110, −8.348328011249800885973387261846, −7.63530036766795514905920417522, −6.59414151088953049182369587127, −5.22332188278719451608421896419, −4.45716615083197105396963816310, −3.75827816894242473562618442266, −1.44094480052751888157058356111,
0.04229658744805105931091864477, 1.69971701327008800451814256544, 2.28769307570449019944993929163, 4.73336083272291837004205451821, 5.56973348780578514927279804565, 6.48960790798164172759303489605, 7.62440648046927550246660670713, 7.889742909101097697979234535045, 8.992110998534309853539943084875, 9.508318989129672651583516354017