L(s) = 1 | + (1.78 + 5.36i)2-s + 29.6i·3-s + (−25.6 + 19.1i)4-s + (−158. + 52.8i)6-s + 90.3·7-s + (−148. − 103. i)8-s − 633.·9-s + 181. i·11-s + (−567. − 758. i)12-s − 455. i·13-s + (161. + 484. i)14-s + (289. − 982. i)16-s + 615.·17-s + (−1.13e3 − 3.39e3i)18-s − 2.49e3i·19-s + ⋯ |
L(s) = 1 | + (0.315 + 0.948i)2-s + 1.89i·3-s + (−0.800 + 0.598i)4-s + (−1.80 + 0.599i)6-s + 0.696·7-s + (−0.821 − 0.570i)8-s − 2.60·9-s + 0.453i·11-s + (−1.13 − 1.52i)12-s − 0.747i·13-s + (0.219 + 0.661i)14-s + (0.282 − 0.959i)16-s + 0.516·17-s + (−0.822 − 2.47i)18-s − 1.58i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.821 + 0.570i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.821 + 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.4737553576\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4737553576\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.78 - 5.36i)T \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - 29.6iT - 243T^{2} \) |
| 7 | \( 1 - 90.3T + 1.68e4T^{2} \) |
| 11 | \( 1 - 181. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 455. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 615.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.49e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 4.42e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.65e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 6.76e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.80e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.88e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 8.26e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 5.72e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.23e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 2.08e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 1.32e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 9.64e3iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 3.30e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.85e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.42e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 8.00e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 8.16e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 2.22e4T + 8.58e9T^{2} \) |
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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
−12.47684876182014469245366780778, −11.28182721227493934640410950959, −10.35040746485606516591445485099, −9.369963533473970808182310396353, −8.610165521063929152187401412167, −7.51129633031039289678654280319, −5.81830486778996244664871343986, −5.01619616709867535592445422720, −4.22464110494665354232187327600, −3.07298722385359718139960286508,
0.12744145511716998631373247800, 1.50705365003963884146911780226, 2.14009628735572284147483914344, 3.73870076719015714356116893303, 5.53728256399413746293677837977, 6.32803161519409592568524409163, 7.86204135901142712257340102165, 8.370979219174774091163080607853, 9.827439828004990414183051921703, 11.23339117053577808702626433278