L(s) = 1 | − 2·2-s + (−3.62 − 3.71i)3-s + 4·4-s + 18.5i·5-s + (7.25 + 7.43i)6-s + 9.26·7-s − 8·8-s + (−0.657 + 26.9i)9-s − 37.1i·10-s + 54.2·11-s + (−14.5 − 14.8i)12-s + 34.8i·13-s − 18.5·14-s + (69.0 − 67.3i)15-s + 16·16-s − 8.98i·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (−0.698 − 0.715i)3-s + 0.5·4-s + 1.66i·5-s + (0.493 + 0.506i)6-s + 0.500·7-s − 0.353·8-s + (−0.0243 + 0.999i)9-s − 1.17i·10-s + 1.48·11-s + (−0.349 − 0.357i)12-s + 0.744i·13-s − 0.353·14-s + (1.18 − 1.15i)15-s + 0.250·16-s − 0.128i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.375 - 0.926i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.375 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.8581767396\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8581767396\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 + (3.62 + 3.71i)T \) |
| 59 | \( 1 + (-419. - 171. i)T \) |
good | 5 | \( 1 - 18.5iT - 125T^{2} \) |
| 7 | \( 1 - 9.26T + 343T^{2} \) |
| 11 | \( 1 - 54.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 34.8iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 8.98iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 123.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 94.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 57.9iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 8.78iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 124. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 316. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 249. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 143.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 716. iT - 1.48e5T^{2} \) |
| 61 | \( 1 + 583. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 11.7iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 14.4iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 1.14e3iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 491.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.08e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.31e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 823. iT - 9.12e5T^{2} \) |
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
−11.27137465033300655209238434196, −10.69066729706855701753526489103, −9.589993542422816204926671725408, −8.412766800488889705696875903541, −7.27833412379203157785636117080, −6.67715985420712409415528074817, −6.08685888699172591340397106767, −4.24676222443643630070998533433, −2.59330726156620027910523242121, −1.44628032440982532167879249832,
0.45432626466161786927561377022, 1.51092871758388778763617561856, 3.85904601005842602281652681329, 4.80905294197961764164915608344, 5.74636751033049471822123168970, 6.87997181047288054370539728698, 8.496366501994139235767483069444, 8.796141409005699992979436051156, 9.725266773870901772075837123556, 10.70030404813547939767397544035