| L(s) = 1 | + (0.642 + 1.26i)2-s + (0.711 + 4.49i)3-s + (−1.17 + 1.61i)4-s + (−5.20 + 3.78i)6-s + (−3.58 + 3.58i)7-s + (−2.79 − 0.442i)8-s + (−11.1 + 3.61i)9-s + (3.53 − 10.8i)11-s + (−8.10 − 4.13i)12-s + (−10.0 + 19.7i)13-s + (−6.82 − 2.21i)14-s + (−1.23 − 3.80i)16-s + (3.10 − 19.6i)17-s + (−11.6 − 11.6i)18-s + (15.8 + 21.8i)19-s + ⋯ |
| L(s) = 1 | + (0.321 + 0.630i)2-s + (0.237 + 1.49i)3-s + (−0.293 + 0.404i)4-s + (−0.867 + 0.630i)6-s + (−0.512 + 0.512i)7-s + (−0.349 − 0.0553i)8-s + (−1.23 + 0.401i)9-s + (0.321 − 0.989i)11-s + (−0.675 − 0.344i)12-s + (−0.773 + 1.51i)13-s + (−0.487 − 0.158i)14-s + (−0.0772 − 0.237i)16-s + (0.182 − 1.15i)17-s + (−0.649 − 0.649i)18-s + (0.833 + 1.14i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 + 0.211i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.164652 - 1.54058i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.164652 - 1.54058i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.642 - 1.26i)T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + (-0.711 - 4.49i)T + (-8.55 + 2.78i)T^{2} \) |
| 7 | \( 1 + (3.58 - 3.58i)T - 49iT^{2} \) |
| 11 | \( 1 + (-3.53 + 10.8i)T + (-97.8 - 71.1i)T^{2} \) |
| 13 | \( 1 + (10.0 - 19.7i)T + (-99.3 - 136. i)T^{2} \) |
| 17 | \( 1 + (-3.10 + 19.6i)T + (-274. - 89.3i)T^{2} \) |
| 19 | \( 1 + (-15.8 - 21.8i)T + (-111. + 343. i)T^{2} \) |
| 23 | \( 1 + (-0.476 + 0.242i)T + (310. - 427. i)T^{2} \) |
| 29 | \( 1 + (3.67 - 5.05i)T + (-259. - 799. i)T^{2} \) |
| 31 | \( 1 + (-5.42 + 3.94i)T + (296. - 913. i)T^{2} \) |
| 37 | \( 1 + (-5.24 - 2.67i)T + (804. + 1.10e3i)T^{2} \) |
| 41 | \( 1 + (-7.33 - 22.5i)T + (-1.35e3 + 988. i)T^{2} \) |
| 43 | \( 1 + (44.7 + 44.7i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-27.2 + 4.31i)T + (2.10e3 - 682. i)T^{2} \) |
| 53 | \( 1 + (-13.6 - 86.0i)T + (-2.67e3 + 868. i)T^{2} \) |
| 59 | \( 1 + (-20.7 + 6.73i)T + (2.81e3 - 2.04e3i)T^{2} \) |
| 61 | \( 1 + (21.3 - 65.6i)T + (-3.01e3 - 2.18e3i)T^{2} \) |
| 67 | \( 1 + (14.4 - 91.1i)T + (-4.26e3 - 1.38e3i)T^{2} \) |
| 71 | \( 1 + (-12.7 - 9.24i)T + (1.55e3 + 4.79e3i)T^{2} \) |
| 73 | \( 1 + (-60.4 + 30.7i)T + (3.13e3 - 4.31e3i)T^{2} \) |
| 79 | \( 1 + (-71.8 + 98.9i)T + (-1.92e3 - 5.93e3i)T^{2} \) |
| 83 | \( 1 + (60.8 + 9.64i)T + (6.55e3 + 2.12e3i)T^{2} \) |
| 89 | \( 1 + (-75.7 - 24.6i)T + (6.40e3 + 4.65e3i)T^{2} \) |
| 97 | \( 1 + (-134. + 21.2i)T + (8.94e3 - 2.90e3i)T^{2} \) |
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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
−12.13799493518163432950436960930, −11.50164173847143471518943360688, −10.12742636586663237552151492104, −9.329067346825192699732754771942, −8.804358600519040808777130785059, −7.35349200038625593707842878036, −6.05690172884541982904897677608, −5.06616259781933635432708366723, −4.01655864577211791286373548407, −2.98184883387258020853132426378,
0.71730035002644883939493364531, 2.16007160748934263692670144529, 3.38905904651428913093499252435, 5.05886477792678136910430943699, 6.42191681468337771218609241837, 7.30616085473680397175437742676, 8.153783494427580137021621790082, 9.577762683139908930649037945521, 10.40616768040272162332674971611, 11.64739415697490287618818515882
