| L(s) = 1 | − 2.82·2-s − 11.3·3-s + 8.00·4-s − 11.1i·5-s + 32.2·6-s + 85.1i·7-s − 22.6·8-s + 48.8·9-s + 31.6i·10-s − 125. i·11-s − 91.1·12-s + 144.·13-s − 240. i·14-s + 127. i·15-s + 64.0·16-s − 52.2i·17-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.26·3-s + 0.500·4-s − 0.447i·5-s + 0.895·6-s + 1.73i·7-s − 0.353·8-s + 0.603·9-s + 0.316i·10-s − 1.03i·11-s − 0.633·12-s + 0.854·13-s − 1.22i·14-s + 0.566i·15-s + 0.250·16-s − 0.180i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.689 - 0.724i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.689 - 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.3659409010\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3659409010\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2.82T \) |
| 5 | \( 1 + 11.1iT \) |
| 23 | \( 1 + (-383. + 364. i)T \) |
| good | 3 | \( 1 + 11.3T + 81T^{2} \) |
| 7 | \( 1 - 85.1iT - 2.40e3T^{2} \) |
| 11 | \( 1 + 125. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 144.T + 2.85e4T^{2} \) |
| 17 | \( 1 + 52.2iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 0.836iT - 1.30e5T^{2} \) |
| 29 | \( 1 + 906.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 1.69e3T + 9.23e5T^{2} \) |
| 37 | \( 1 - 1.69e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + 2.58e3T + 2.82e6T^{2} \) |
| 43 | \( 1 - 364. iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 2.59e3T + 4.87e6T^{2} \) |
| 53 | \( 1 - 1.53e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 1.52e3T + 1.21e7T^{2} \) |
| 61 | \( 1 - 5.32e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 4.91e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 1.79e3T + 2.54e7T^{2} \) |
| 73 | \( 1 - 607.T + 2.83e7T^{2} \) |
| 79 | \( 1 - 229. iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 6.88e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 3.67e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.03e4iT - 8.85e7T^{2} \) |
| 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
−11.63175420872318386497672247656, −11.21757730400318536070878971462, −10.00907508908490402121658368249, −8.759368026716992901459912411369, −8.385659573820979940213742247644, −6.50825393216015999063524492756, −5.88115294272760146927656922608, −5.01096922110832775813843071130, −2.90386253652185959613930411426, −1.16680661735648385520646180945,
0.22178539726403326478295653766, 1.44508348097688247113518718018, 3.64127323290830062347493275041, 4.94943734541346124982488894591, 6.40195829854079773521644401931, 7.01084783358697579928839020492, 7.968212697648452224388151546724, 9.596380741463129663350029330326, 10.44574221753447744137408817028, 10.97355953185036148169976028420
