| L(s) = 1 | + (−3.55 + 0.626i)2-s + (−2.31 − 2.76i)3-s + (8.45 − 3.07i)4-s + (−3.75 − 1.36i)5-s + (9.95 + 8.35i)6-s + (2.5 − 4.33i)7-s + (−15.6 + 9.01i)8-s + (−0.694 + 3.93i)9-s + (14.2 + 2.50i)10-s + (5 + 8.66i)11-s + (−28.1 − 16.2i)12-s + (−2.31 + 2.76i)13-s + (−6.16 + 16.9i)14-s + (4.93 + 13.5i)15-s + (22.2 − 18.6i)16-s + (2.60 + 14.7i)17-s + ⋯ |
| L(s) = 1 | + (−1.77 + 0.313i)2-s + (−0.772 − 0.920i)3-s + (2.11 − 0.769i)4-s + (−0.751 − 0.273i)5-s + (1.65 + 1.39i)6-s + (0.357 − 0.618i)7-s + (−1.95 + 1.12i)8-s + (−0.0771 + 0.437i)9-s + (1.42 + 0.250i)10-s + (0.454 + 0.787i)11-s + (−2.34 − 1.35i)12-s + (−0.178 + 0.212i)13-s + (−0.440 + 1.21i)14-s + (0.328 + 0.903i)15-s + (1.38 − 1.16i)16-s + (0.153 + 0.868i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0158i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0158i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.375712 + 0.00296924i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.375712 + 0.00296924i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 \) |
| good | 2 | \( 1 + (3.55 - 0.626i)T + (3.75 - 1.36i)T^{2} \) |
| 3 | \( 1 + (2.31 + 2.76i)T + (-1.56 + 8.86i)T^{2} \) |
| 5 | \( 1 + (3.75 + 1.36i)T + (19.1 + 16.0i)T^{2} \) |
| 7 | \( 1 + (-2.5 + 4.33i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-5 - 8.66i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (2.31 - 2.76i)T + (-29.3 - 166. i)T^{2} \) |
| 17 | \( 1 + (-2.60 - 14.7i)T + (-271. + 98.8i)T^{2} \) |
| 23 | \( 1 + (32.8 - 11.9i)T + (405. - 340. i)T^{2} \) |
| 29 | \( 1 + (-17.7 - 3.13i)T + (790. + 287. i)T^{2} \) |
| 31 | \( 1 + (-31.2 - 18.0i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 21.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-23.1 - 27.6i)T + (-291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (-18.7 - 6.84i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-1.73 + 9.84i)T + (-2.07e3 - 755. i)T^{2} \) |
| 53 | \( 1 + (-25.8 - 71.1i)T + (-2.15e3 + 1.80e3i)T^{2} \) |
| 59 | \( 1 + (17.7 - 3.13i)T + (3.27e3 - 1.19e3i)T^{2} \) |
| 61 | \( 1 + (-37.5 + 13.6i)T + (2.85e3 - 2.39e3i)T^{2} \) |
| 67 | \( 1 + (-39.0 - 6.88i)T + (4.21e3 + 1.53e3i)T^{2} \) |
| 71 | \( 1 + (-36.9 + 101. i)T + (-3.86e3 - 3.24e3i)T^{2} \) |
| 73 | \( 1 + (-80.4 + 67.4i)T + (925. - 5.24e3i)T^{2} \) |
| 79 | \( 1 + (23.1 + 27.6i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-20 + 34.6i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-1.37e3 - 7.80e3i)T^{2} \) |
| 97 | \( 1 + (120. - 21.2i)T + (8.84e3 - 3.21e3i)T^{2} \) |
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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
−11.08898326487227241087285844634, −10.25384837292172322904306588395, −9.327347603495739347527930451483, −8.098245043300982867508592577249, −7.66287409722471980872481548121, −6.79782787591945429076697611657, −6.01723907285396094596667954506, −4.23189845979763169556255179962, −1.83023287763046781554958588655, −0.813896690291893913986375839763,
0.51127273201889242354733427163, 2.48777447692324057226213421103, 3.97988402162834109262188521847, 5.49068447505666009368265737163, 6.66627450236755769639929603901, 7.897785769784976119616969144879, 8.473763428094796007867116094868, 9.604409980078278304055281682623, 10.18463447399206109037694350721, 11.16336936652675826533060878698
