| L(s) = 1 | + 2.82i·2-s + (8.93 − 1.10i)3-s − 8.00·4-s − 7.22i·5-s + (3.11 + 25.2i)6-s + 73.4·7-s − 22.6i·8-s + (78.5 − 19.6i)9-s + 20.4·10-s − 129. i·11-s + (−71.4 + 8.81i)12-s − 211.·13-s + 207. i·14-s + (−7.95 − 64.5i)15-s + 64.0·16-s − 427. i·17-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + (0.992 − 0.122i)3-s − 0.500·4-s − 0.288i·5-s + (0.0865 + 0.701i)6-s + 1.49·7-s − 0.353i·8-s + (0.970 − 0.242i)9-s + 0.204·10-s − 1.07i·11-s + (−0.496 + 0.0612i)12-s − 1.24·13-s + 1.06i·14-s + (−0.0353 − 0.286i)15-s + 0.250·16-s − 1.47i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 174 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.122i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 174 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.992 - 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.878893515\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.878893515\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2.82iT \) |
| 3 | \( 1 + (-8.93 + 1.10i)T \) |
| 29 | \( 1 + 156. iT \) |
| good | 5 | \( 1 + 7.22iT - 625T^{2} \) |
| 7 | \( 1 - 73.4T + 2.40e3T^{2} \) |
| 11 | \( 1 + 129. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 211.T + 2.85e4T^{2} \) |
| 17 | \( 1 + 427. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 362.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 682. iT - 2.79e5T^{2} \) |
| 31 | \( 1 + 638.T + 9.23e5T^{2} \) |
| 37 | \( 1 - 438.T + 1.87e6T^{2} \) |
| 41 | \( 1 - 311. iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 2.71e3T + 3.41e6T^{2} \) |
| 47 | \( 1 - 3.42e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 2.09e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 4.92e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 4.85e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 1.50e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 3.44e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 6.34e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 1.80e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 9.05e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 1.40e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.05e4T + 8.85e7T^{2} \) |
| 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
−12.15474619261103192272413819820, −11.13994617017434521498743553659, −9.581614003286306877644515684279, −8.894807131013713729062457621089, −7.73841475639834021486998243210, −7.35999338662910931909633080276, −5.43995275310480668044830260716, −4.54601825869828116874881985238, −2.88491730075364892495620413895, −1.12507778956927697102902601039,
1.60015721661885313631006302185, 2.55590096113400922139515863148, 4.16796335568178819157408191143, 5.01596645772876082836488326715, 7.20302119858120553273799300592, 8.033721594512071083306214937074, 9.037877114082018343691945410931, 10.13751990762337027043058086838, 10.83749092005275045982950292335, 12.16770306471672628415852641559
