L(s) = 1 | + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (2.23 − 0.133i)5-s + (−0.866 − 2.5i)7-s − 0.999i·8-s + (1.86 − 1.23i)10-s − 2i·13-s + (−2 − 1.73i)14-s + (−0.5 − 0.866i)16-s + (1.73 + i)17-s + (−1 − 1.73i)19-s + (0.999 − 1.99i)20-s + (0.866 − 0.5i)23-s + (4.96 − 0.598i)25-s + (−1 − 1.73i)26-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (0.998 − 0.0599i)5-s + (−0.327 − 0.944i)7-s − 0.353i·8-s + (0.590 − 0.389i)10-s − 0.554i·13-s + (−0.534 − 0.462i)14-s + (−0.125 − 0.216i)16-s + (0.420 + 0.242i)17-s + (−0.229 − 0.397i)19-s + (0.223 − 0.447i)20-s + (0.180 − 0.104i)23-s + (0.992 − 0.119i)25-s + (−0.196 − 0.339i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.324 + 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.324 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.91649 - 1.36928i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.91649 - 1.36928i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.23 + 0.133i)T \) |
| 7 | \( 1 + (0.866 + 2.5i)T \) |
good | 11 | \( 1 + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + (-1.73 - i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.866 + 0.5i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + T + 29T^{2} \) |
| 31 | \( 1 + (5 - 8.66i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.92 + 4i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 - 5iT - 43T^{2} \) |
| 47 | \( 1 + (-6.92 + 4i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (5.19 + 3i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (1 - 1.73i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.5 - 7.79i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.06 + 3.5i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + (-8.66 - 5i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5 + 8.66i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 9iT - 83T^{2} \) |
| 89 | \( 1 + (-3.5 - 6.06i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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
−10.53216214462286160476356729339, −9.783745975686101809926788947500, −8.917905033063397482896407686805, −7.58530134593419210693672132332, −6.67283762913732135692701177291, −5.79290821752424911697524805141, −4.88042480040498311830414139625, −3.72754148573961795538465383326, −2.62652323393542724311638625983, −1.17532756090815000519104581651,
1.97676035651480685141837753913, 2.99167260709427651732464272674, 4.36344574377281478588780753210, 5.57851149995383191972348638540, 6.00267055428814973770936797386, 6.98352900191776595433218816888, 8.085274543952564062568800201166, 9.198836590801732566440159019566, 9.658075549004584543433779415784, 10.86496704420777038541166655435