| L(s) = 1 | − 1.38·2-s − 6.08·4-s − 7.07i·5-s + 14.2i·7-s + 19.4·8-s + 9.78i·10-s + (36.4 + 1.29i)11-s + 78.5i·13-s − 19.6i·14-s + 21.7·16-s + 15.8·17-s + 32.0i·19-s + 43.0i·20-s + (−50.4 − 1.78i)22-s + 114. i·23-s + ⋯ |
| L(s) = 1 | − 0.489·2-s − 0.760·4-s − 0.632i·5-s + 0.768i·7-s + 0.861·8-s + 0.309i·10-s + (0.999 + 0.0354i)11-s + 1.67i·13-s − 0.375i·14-s + 0.339·16-s + 0.226·17-s + 0.386i·19-s + 0.481i·20-s + (−0.488 − 0.0173i)22-s + 1.03i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.865516 + 0.428767i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.865516 + 0.428767i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + (-36.4 - 1.29i)T \) |
| good | 2 | \( 1 + 1.38T + 8T^{2} \) |
| 5 | \( 1 + 7.07iT - 125T^{2} \) |
| 7 | \( 1 - 14.2iT - 343T^{2} \) |
| 13 | \( 1 - 78.5iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 15.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 32.0iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 114. iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 114.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 129.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 301.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 287.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 105. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 240. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 320. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 600. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 398. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 260T + 3.00e5T^{2} \) |
| 71 | \( 1 - 672. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 1.14e3iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 899. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 531.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.19e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 404.T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.69407025722280879753317268292, −12.36291292294447998530660650537, −11.63378609945540044548462439090, −9.965504300609467763005987686270, −9.042472537610975155852882691952, −8.506726362441253024385119662359, −6.84147949375082228334198830123, −5.23965957533741317478846544450, −4.01892062113478652049515215829, −1.47432702258862953238167277875,
0.77073262116733273510402524057, 3.37046151588962614975612165024, 4.83365329074903548123375771090, 6.57046475581154235682910469220, 7.79722667102222646735740429245, 8.843101839179723912473901543834, 10.20468012828258079229951892170, 10.65294765595385146164420829286, 12.25781668507498470550048791646, 13.40212534539279075036277156135
