L(s) = 1 | + (0.866 + 0.5i)2-s + (1.09 − 0.294i)3-s + (0.499 + 0.866i)4-s + (0.803 + 2.08i)5-s + (1.09 + 0.294i)6-s + (−2.54 − 4.40i)7-s + 0.999i·8-s + (−1.47 + 0.854i)9-s + (−0.347 + 2.20i)10-s + (−2.27 + 0.609i)11-s + (0.803 + 0.803i)12-s + (3.54 − 0.653i)13-s − 5.08i·14-s + (1.49 + 2.05i)15-s + (−0.5 + 0.866i)16-s + (−0.0601 + 0.224i)17-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (0.633 − 0.169i)3-s + (0.249 + 0.433i)4-s + (0.359 + 0.933i)5-s + (0.448 + 0.120i)6-s + (−0.961 − 1.66i)7-s + 0.353i·8-s + (−0.493 + 0.284i)9-s + (−0.109 + 0.698i)10-s + (−0.685 + 0.183i)11-s + (0.231 + 0.231i)12-s + (0.983 − 0.181i)13-s − 1.35i·14-s + (0.386 + 0.530i)15-s + (−0.125 + 0.216i)16-s + (−0.0145 + 0.0544i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.886 - 0.462i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.886 - 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.58060 + 0.387809i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.58060 + 0.387809i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 5 | \( 1 + (-0.803 - 2.08i)T \) |
| 13 | \( 1 + (-3.54 + 0.653i)T \) |
good | 3 | \( 1 + (-1.09 + 0.294i)T + (2.59 - 1.5i)T^{2} \) |
| 7 | \( 1 + (2.54 + 4.40i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.27 - 0.609i)T + (9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 + (0.0601 - 0.224i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-1.53 + 5.71i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-0.674 - 2.51i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (2.64 + 1.52i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (4.45 - 4.45i)T - 31iT^{2} \) |
| 37 | \( 1 + (-2.49 + 4.31i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.412 + 1.53i)T + (-35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (-1.66 - 0.447i)T + (37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 - 12.2T + 47T^{2} \) |
| 53 | \( 1 + (-5.79 - 5.79i)T + 53iT^{2} \) |
| 59 | \( 1 + (2.68 + 0.720i)T + (51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (-4.23 - 7.33i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.150 - 0.0866i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.17 - 1.11i)T + (61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + 7.51iT - 73T^{2} \) |
| 79 | \( 1 + 5.17iT - 79T^{2} \) |
| 83 | \( 1 + 10.4T + 83T^{2} \) |
| 89 | \( 1 + (-1.49 - 5.59i)T + (-77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (7.02 - 4.05i)T + (48.5 - 84.0i)T^{2} \) |
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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
−13.59673160144605244833110390181, −12.99740057871187687299675670481, −11.10305822302115922443893402330, −10.52700787279612328698549312873, −9.170898195401335794731381575764, −7.57129780219018880010947535711, −7.02614059262635501161455266155, −5.72477393590659082926599190217, −3.82513555513238304471628745603, −2.83214390743676487075129984002,
2.37304748355670007401130697669, 3.64026304453291543733098431316, 5.51994418430440175992920374452, 6.04979889500919811423842239526, 8.322683568142522724767850059840, 9.055975842286224243405728905564, 9.901454574393125541814435260941, 11.50924147289549130490781428863, 12.46053952825172938381273692989, 13.08133194455526302388324256140