L(s) = 1 | + (−1.36 − 0.366i)2-s + (−0.866 − 1.5i)3-s + (1.73 + i)4-s + (2.23 − 0.133i)5-s + (0.633 + 2.36i)6-s + (−0.767 − 2.86i)7-s + (−1.99 − 2i)8-s + (−1.5 + 2.59i)9-s + (−3.09 − 0.633i)10-s + (1 − 1.73i)11-s − 3.46i·12-s + (6.46 + 1.73i)13-s + 4.19i·14-s + (−2.13 − 3.23i)15-s + (1.99 + 3.46i)16-s + (−3 − 3i)17-s + ⋯ |
L(s) = 1 | + (−0.965 − 0.258i)2-s + (−0.499 − 0.866i)3-s + (0.866 + 0.5i)4-s + (0.998 − 0.0599i)5-s + (0.258 + 0.965i)6-s + (−0.290 − 1.08i)7-s + (−0.707 − 0.707i)8-s + (−0.5 + 0.866i)9-s + (−0.979 − 0.200i)10-s + (0.301 − 0.522i)11-s − 0.999i·12-s + (1.79 + 0.480i)13-s + 1.12i·14-s + (−0.550 − 0.834i)15-s + (0.499 + 0.866i)16-s + (−0.727 − 0.727i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.395 + 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.395 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.460744 - 0.699851i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.460744 - 0.699851i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.36 + 0.366i)T \) |
| 3 | \( 1 + (0.866 + 1.5i)T \) |
| 5 | \( 1 + (-2.23 + 0.133i)T \) |
good | 7 | \( 1 + (0.767 + 2.86i)T + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-6.46 - 1.73i)T + (11.2 + 6.5i)T^{2} \) |
| 17 | \( 1 + (3 + 3i)T + 17iT^{2} \) |
| 19 | \( 1 - 1.26T + 19T^{2} \) |
| 23 | \( 1 + (3.23 + 0.866i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (6.23 + 3.59i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (1.09 + 1.90i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (6.46 + 6.46i)T + 37iT^{2} \) |
| 41 | \( 1 + (0.401 - 0.232i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.169 - 0.633i)T + (-37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + (5.59 - 1.5i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-5.19 - 5.19i)T + 53iT^{2} \) |
| 59 | \( 1 + (-10.5 + 6.09i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.30 - 1.33i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.33 - 4.96i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 12iT - 71T^{2} \) |
| 73 | \( 1 + (-0.196 - 0.196i)T + 73iT^{2} \) |
| 79 | \( 1 + (7.09 + 4.09i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-9.96 + 2.66i)T + (71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 - 14.6T + 89T^{2} \) |
| 97 | \( 1 + (-1.83 - 6.83i)T + (-84.0 + 48.5i)T^{2} \) |
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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
−11.07198591118926751874409518798, −10.37680321832291065054379823664, −9.270084708931484413999817676490, −8.460760084699498731497810341785, −7.26842937255263520728776205640, −6.52743522837608655609495795542, −5.78363202728331103823639317830, −3.75227504656567874975225134702, −2.06699578117492395839295005303, −0.870115731589868472600962408350,
1.78185639619690260800517351482, 3.40554441284297033781978645065, 5.31956495790572260421940682472, 5.99253964535360062305549473134, 6.68123091850912379968081682033, 8.537433894834538864623849870231, 8.972190271443633507815261344918, 9.858428825795969717227450454290, 10.58701730159125283147090526399, 11.36183602088592094932999036061