L(s) = 1 | + (−0.155 − 0.0897i)2-s + (2.14 − 4.73i)3-s + (−3.98 − 6.90i)4-s + (2.5 − 4.33i)5-s + (−0.757 + 0.544i)6-s + (18.1 + 3.54i)7-s + 2.86i·8-s + (−17.8 − 20.2i)9-s + (−0.777 + 0.448i)10-s + (−42.8 + 24.7i)11-s + (−41.1 + 4.09i)12-s − 62.5i·13-s + (−2.50 − 2.18i)14-s + (−15.1 − 21.1i)15-s + (−31.6 + 54.7i)16-s + (−19.4 − 33.6i)17-s + ⋯ |
L(s) = 1 | + (−0.0549 − 0.0317i)2-s + (0.411 − 0.911i)3-s + (−0.497 − 0.862i)4-s + (0.223 − 0.387i)5-s + (−0.0515 + 0.0370i)6-s + (0.981 + 0.191i)7-s + 0.126i·8-s + (−0.660 − 0.750i)9-s + (−0.0245 + 0.0141i)10-s + (−1.17 + 0.677i)11-s + (−0.991 + 0.0985i)12-s − 1.33i·13-s + (−0.0478 − 0.0416i)14-s + (−0.260 − 0.363i)15-s + (−0.493 + 0.855i)16-s + (−0.277 − 0.480i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.677 + 0.735i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.677 + 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.596517 - 1.35970i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.596517 - 1.35970i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-2.14 + 4.73i)T \) |
| 5 | \( 1 + (-2.5 + 4.33i)T \) |
| 7 | \( 1 + (-18.1 - 3.54i)T \) |
good | 2 | \( 1 + (0.155 + 0.0897i)T + (4 + 6.92i)T^{2} \) |
| 11 | \( 1 + (42.8 - 24.7i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + 62.5iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (19.4 + 33.6i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-14.1 - 8.18i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-125. - 72.6i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + 246. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (-128. + 74.0i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (174. - 301. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 429.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 73.5T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-124. + 214. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-263. + 152. i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-336. - 582. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (279. + 161. i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (103. + 179. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 717. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-84.3 + 48.7i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-450. + 779. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 90.9T + 5.71e5T^{2} \) |
| 89 | \( 1 + (328. - 568. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 1.46e3iT - 9.12e5T^{2} \) |
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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
−13.23938677331210636450114354949, −11.99278266032589990393415520435, −10.71141046885402964835992827085, −9.571655733066467231140837613530, −8.388045017707559810973687152350, −7.54638617570111769712101015554, −5.78959400127588273647579372476, −4.88783867796158715435379011831, −2.39505474376889468083533498737, −0.845437086367990935928644352732,
2.68914496566094047693962371457, 4.13205718760903849581528883042, 5.19942300455122844277553589704, 7.26270012509889330371166771034, 8.443696359167075447372160578586, 9.103047538537615378527128310566, 10.61570949522129224583536376358, 11.24962114960012470042404929723, 12.77021096823349910958576252262, 13.96702314663094957676109665331