L(s) = 1 | + (0.707 − 1.70i)3-s + (3.12 − 1.29i)5-s + (−1 + i)7-s + (−0.292 − 0.292i)9-s + (0.121 + 0.292i)11-s + (−1.70 − 0.707i)13-s − 6.24i·15-s + 2.82i·17-s + (−5.53 − 2.29i)19-s + (0.999 + 2.41i)21-s + (−0.171 − 0.171i)23-s + (4.53 − 4.53i)25-s + (4.41 − 1.82i)27-s + (−1.12 + 2.70i)29-s + 4·31-s + ⋯ |
L(s) = 1 | + (0.408 − 0.985i)3-s + (1.39 − 0.578i)5-s + (−0.377 + 0.377i)7-s + (−0.0976 − 0.0976i)9-s + (0.0365 + 0.0883i)11-s + (−0.473 − 0.196i)13-s − 1.61i·15-s + 0.685i·17-s + (−1.26 − 0.526i)19-s + (0.218 + 0.526i)21-s + (−0.0357 − 0.0357i)23-s + (0.907 − 0.907i)25-s + (0.849 − 0.351i)27-s + (−0.208 + 0.502i)29-s + 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.45245 - 0.776355i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.45245 - 0.776355i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (-0.707 + 1.70i)T + (-2.12 - 2.12i)T^{2} \) |
| 5 | \( 1 + (-3.12 + 1.29i)T + (3.53 - 3.53i)T^{2} \) |
| 7 | \( 1 + (1 - i)T - 7iT^{2} \) |
| 11 | \( 1 + (-0.121 - 0.292i)T + (-7.77 + 7.77i)T^{2} \) |
| 13 | \( 1 + (1.70 + 0.707i)T + (9.19 + 9.19i)T^{2} \) |
| 17 | \( 1 - 2.82iT - 17T^{2} \) |
| 19 | \( 1 + (5.53 + 2.29i)T + (13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (0.171 + 0.171i)T + 23iT^{2} \) |
| 29 | \( 1 + (1.12 - 2.70i)T + (-20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + (1.70 - 0.707i)T + (26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (5.82 + 5.82i)T + 41iT^{2} \) |
| 43 | \( 1 + (-3.29 - 7.94i)T + (-30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 - 11.6iT - 47T^{2} \) |
| 53 | \( 1 + (3.12 + 7.53i)T + (-37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (6.12 - 2.53i)T + (41.7 - 41.7i)T^{2} \) |
| 61 | \( 1 + (0.292 - 0.707i)T + (-43.1 - 43.1i)T^{2} \) |
| 67 | \( 1 + (-1.53 + 3.70i)T + (-47.3 - 47.3i)T^{2} \) |
| 71 | \( 1 + (-0.171 + 0.171i)T - 71iT^{2} \) |
| 73 | \( 1 + (-7 - 7i)T + 73iT^{2} \) |
| 79 | \( 1 + 6iT - 79T^{2} \) |
| 83 | \( 1 + (-6.12 - 2.53i)T + (58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (2.65 - 2.65i)T - 89iT^{2} \) |
| 97 | \( 1 + 1.51T + 97T^{2} \) |
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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
−12.48111179317740827082437101154, −10.79514573516505097937504853930, −9.806194492846366324785121846294, −8.940556600901434289952505883115, −8.046690377839520544090238194354, −6.73647444510445565589765587313, −5.99248497263894253253794658931, −4.73713303565563725836505926758, −2.63129916045971228290448228239, −1.62456651384583248660789909657,
2.27388619249460429245900498453, 3.57789961753940801480259924595, 4.84931891560977910494127252159, 6.14019827252509520796946122644, 7.01486095300628755445851119949, 8.612489646639570121176308646373, 9.619539174654769335542201518326, 10.06998874607469122729459827728, 10.77453010729469106516727179332, 12.18702740014711959242028943035