L(s) = 1 | − i·2-s + (−1.69 − 0.345i)3-s − 4-s + (−0.866 + 0.5i)5-s + (−0.345 + 1.69i)6-s + (0.742 − 1.28i)7-s + i·8-s + (2.76 + 1.17i)9-s + (0.5 + 0.866i)10-s + (−2.14 − 3.71i)11-s + (1.69 + 0.345i)12-s + (−1.46 + 0.848i)13-s + (−1.28 − 0.742i)14-s + (1.64 − 0.549i)15-s + 16-s + (−2.87 + 4.97i)17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (−0.979 − 0.199i)3-s − 0.5·4-s + (−0.387 + 0.223i)5-s + (−0.141 + 0.692i)6-s + (0.280 − 0.486i)7-s + 0.353i·8-s + (0.920 + 0.391i)9-s + (0.158 + 0.273i)10-s + (−0.645 − 1.11i)11-s + (0.489 + 0.0998i)12-s + (−0.407 + 0.235i)13-s + (−0.343 − 0.198i)14-s + (0.424 − 0.141i)15-s + 0.250·16-s + (−0.697 + 1.20i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.791 - 0.611i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.791 - 0.611i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.501645 + 0.171331i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.501645 + 0.171331i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + (1.69 + 0.345i)T \) |
| 5 | \( 1 + (0.866 - 0.5i)T \) |
| 31 | \( 1 + (-2.50 + 4.97i)T \) |
good | 7 | \( 1 + (-0.742 + 1.28i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.14 + 3.71i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.46 - 0.848i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (2.87 - 4.97i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.59 - 2.77i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 4.16T + 23T^{2} \) |
| 29 | \( 1 - 7.18T + 29T^{2} \) |
| 37 | \( 1 + (-8.23 - 4.75i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (9.76 - 5.63i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-6.44 - 3.72i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 11.9iT - 47T^{2} \) |
| 53 | \( 1 + (-3.03 - 5.25i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.18 - 0.682i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + 12.5iT - 61T^{2} \) |
| 67 | \( 1 + (-5.98 - 10.3i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (0.650 - 0.375i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (3.41 - 1.96i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-9.64 - 5.56i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (4.88 + 8.46i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 4.38T + 89T^{2} \) |
| 97 | \( 1 + 14.9T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.47095557661673271274239593789, −9.693825945942707753852498027719, −8.205412806172868587192913049005, −7.919055150757200708554240434917, −6.49742155412304096323144162582, −5.91092171741729750200600497230, −4.62783153471550613193648546281, −4.02787583875104451861235375013, −2.59957902501973378646476715518, −1.14450758286537946278794544404,
0.32814493080095603466527817990, 2.35718657482055459505273970231, 4.15595394726022437623593429082, 4.93923758479717170952389705794, 5.40902114556695773460532351405, 6.72306765779545532955522107126, 7.17981382997464171889678763942, 8.207340712201884300271734198760, 9.120260939425282498642812004972, 10.01054618130853053145473064857