| L(s) = 1 | + (0.434 + 0.502i)3-s + (1.14 − 0.165i)5-s + (−11.0 − 5.04i)7-s + (1.21 − 8.47i)9-s + (−4.36 + 14.8i)11-s + (6.62 + 14.4i)13-s + (0.582 + 0.504i)15-s + (−15.4 + 24.1i)17-s + (8.54 + 13.3i)19-s + (−2.27 − 7.74i)21-s + (−19.5 + 12.0i)23-s + (−22.6 + 6.66i)25-s + (9.81 − 6.30i)27-s + (−13.2 − 8.50i)29-s + (28.8 − 33.3i)31-s + ⋯ |
| L(s) = 1 | + (0.144 + 0.167i)3-s + (0.229 − 0.0330i)5-s + (−1.57 − 0.721i)7-s + (0.135 − 0.941i)9-s + (−0.396 + 1.35i)11-s + (0.509 + 1.11i)13-s + (0.0388 + 0.0336i)15-s + (−0.911 + 1.41i)17-s + (0.449 + 0.700i)19-s + (−0.108 − 0.368i)21-s + (−0.851 + 0.523i)23-s + (−0.907 + 0.266i)25-s + (0.363 − 0.233i)27-s + (−0.456 − 0.293i)29-s + (0.931 − 1.07i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.730 - 0.682i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.730 - 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.228797 + 0.579955i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.228797 + 0.579955i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 + (19.5 - 12.0i)T \) |
| good | 3 | \( 1 + (-0.434 - 0.502i)T + (-1.28 + 8.90i)T^{2} \) |
| 5 | \( 1 + (-1.14 + 0.165i)T + (23.9 - 7.04i)T^{2} \) |
| 7 | \( 1 + (11.0 + 5.04i)T + (32.0 + 37.0i)T^{2} \) |
| 11 | \( 1 + (4.36 - 14.8i)T + (-101. - 65.4i)T^{2} \) |
| 13 | \( 1 + (-6.62 - 14.4i)T + (-110. + 127. i)T^{2} \) |
| 17 | \( 1 + (15.4 - 24.1i)T + (-120. - 262. i)T^{2} \) |
| 19 | \( 1 + (-8.54 - 13.3i)T + (-149. + 328. i)T^{2} \) |
| 29 | \( 1 + (13.2 + 8.50i)T + (349. + 765. i)T^{2} \) |
| 31 | \( 1 + (-28.8 + 33.3i)T + (-136. - 951. i)T^{2} \) |
| 37 | \( 1 + (29.8 + 4.29i)T + (1.31e3 + 385. i)T^{2} \) |
| 41 | \( 1 + (2.13 + 14.8i)T + (-1.61e3 + 473. i)T^{2} \) |
| 43 | \( 1 + (35.9 - 31.1i)T + (263. - 1.83e3i)T^{2} \) |
| 47 | \( 1 - 17.3T + 2.20e3T^{2} \) |
| 53 | \( 1 + (40.8 + 18.6i)T + (1.83e3 + 2.12e3i)T^{2} \) |
| 59 | \( 1 + (-0.787 - 1.72i)T + (-2.27e3 + 2.63e3i)T^{2} \) |
| 61 | \( 1 + (-41.8 - 36.3i)T + (529. + 3.68e3i)T^{2} \) |
| 67 | \( 1 + (17.0 + 58.2i)T + (-3.77e3 + 2.42e3i)T^{2} \) |
| 71 | \( 1 + (82.0 - 24.0i)T + (4.24e3 - 2.72e3i)T^{2} \) |
| 73 | \( 1 + (-13.8 + 8.89i)T + (2.21e3 - 4.84e3i)T^{2} \) |
| 79 | \( 1 + (-95.6 + 43.6i)T + (4.08e3 - 4.71e3i)T^{2} \) |
| 83 | \( 1 + (-61.2 - 8.79i)T + (6.60e3 + 1.94e3i)T^{2} \) |
| 89 | \( 1 + (62.4 - 54.1i)T + (1.12e3 - 7.84e3i)T^{2} \) |
| 97 | \( 1 + (10.9 - 1.57i)T + (9.02e3 - 2.65e3i)T^{2} \) |
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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
−11.64097532852404057574302953707, −10.20520713906438986755507894436, −9.828978598490362551757230102088, −9.073171773965153051133281019064, −7.67657239336024035732402454221, −6.60570947244518641446101145292, −6.09642454108215653746313494064, −4.19263408407140859724102594459, −3.64958697008007807350772440315, −1.87114466252536977869579818392,
0.25724699558448075017756525874, 2.56573263827916943541454789439, 3.27477337681916210761947654399, 5.13201621513758082082215635309, 5.98219657348955984149181348015, 6.92185639346602161051560464001, 8.187456934928440778332941520258, 8.934221384592513794619906553823, 9.958148125758401450701987453231, 10.73853781548165869918501881669
