| L(s) = 1 | + (−1.39 − 0.242i)2-s + (1.17 − 1.33i)3-s + (1.88 + 0.676i)4-s + (−2.65 − 0.174i)5-s + (−1.95 + 1.57i)6-s + (−0.123 + 2.64i)7-s + (−2.45 − 1.39i)8-s + (−0.0196 − 0.149i)9-s + (3.65 + 0.887i)10-s + (1.51 + 0.512i)11-s + (3.10 − 1.72i)12-s + (1.63 + 2.45i)13-s + (0.813 − 3.65i)14-s + (−3.34 + 3.34i)15-s + (3.08 + 2.54i)16-s + (6.79 − 1.81i)17-s + ⋯ |
| L(s) = 1 | + (−0.985 − 0.171i)2-s + (0.675 − 0.770i)3-s + (0.941 + 0.338i)4-s + (−1.18 − 0.0778i)5-s + (−0.797 + 0.643i)6-s + (−0.0466 + 0.998i)7-s + (−0.869 − 0.494i)8-s + (−0.00654 − 0.0497i)9-s + (1.15 + 0.280i)10-s + (0.455 + 0.154i)11-s + (0.896 − 0.496i)12-s + (0.454 + 0.679i)13-s + (0.217 − 0.976i)14-s + (−0.862 + 0.862i)15-s + (0.771 + 0.636i)16-s + (1.64 − 0.441i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0658i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0658i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.975896 + 0.0321468i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.975896 + 0.0321468i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.39 + 0.242i)T \) |
| 7 | \( 1 + (0.123 - 2.64i)T \) |
| good | 3 | \( 1 + (-1.17 + 1.33i)T + (-0.391 - 2.97i)T^{2} \) |
| 5 | \( 1 + (2.65 + 0.174i)T + (4.95 + 0.652i)T^{2} \) |
| 11 | \( 1 + (-1.51 - 0.512i)T + (8.72 + 6.69i)T^{2} \) |
| 13 | \( 1 + (-1.63 - 2.45i)T + (-4.97 + 12.0i)T^{2} \) |
| 17 | \( 1 + (-6.79 + 1.81i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-4.66 - 2.30i)T + (11.5 + 15.0i)T^{2} \) |
| 23 | \( 1 + (1.52 - 0.200i)T + (22.2 - 5.95i)T^{2} \) |
| 29 | \( 1 + (-1.30 + 6.58i)T + (-26.7 - 11.0i)T^{2} \) |
| 31 | \( 1 + (5.28 + 3.05i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.0575 - 0.877i)T + (-36.6 - 4.82i)T^{2} \) |
| 41 | \( 1 + (4.82 - 11.6i)T + (-28.9 - 28.9i)T^{2} \) |
| 43 | \( 1 + (-3.59 + 0.716i)T + (39.7 - 16.4i)T^{2} \) |
| 47 | \( 1 + (-0.310 + 1.15i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-10.5 - 3.59i)T + (42.0 + 32.2i)T^{2} \) |
| 59 | \( 1 + (0.926 + 1.87i)T + (-35.9 + 46.8i)T^{2} \) |
| 61 | \( 1 + (-4.15 - 12.2i)T + (-48.3 + 37.1i)T^{2} \) |
| 67 | \( 1 + (-4.30 + 4.90i)T + (-8.74 - 66.4i)T^{2} \) |
| 71 | \( 1 + (1.19 - 0.494i)T + (50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (2.08 - 1.60i)T + (18.8 - 70.5i)T^{2} \) |
| 79 | \( 1 + (-8.19 - 2.19i)T + (68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (-3.27 + 2.18i)T + (31.7 - 76.6i)T^{2} \) |
| 89 | \( 1 + (9.75 - 12.7i)T + (-23.0 - 85.9i)T^{2} \) |
| 97 | \( 1 + 3.71iT - 97T^{2} \) |
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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
−11.39155882233724641954172933869, −9.932614542094720168433390746334, −9.169509369836550969592703992670, −8.134771765622750553174263037319, −7.84514769125564619102156933751, −6.92820590296790761038744872619, −5.67714996807984978201912756854, −3.78971469958150412713306817426, −2.71769022653823744694466670298, −1.36613278138010790896250556928,
0.929370921990951116609417407240, 3.38044477834467952977168438239, 3.67892248963555582323182761578, 5.38692393929126466711343695251, 6.91164098907578818466444752125, 7.62897189079302400889141392042, 8.385159385015222792462760603176, 9.235755453079086235918343244703, 10.18950518913813694159168771687, 10.71935384399668690800044304362
