L(s) = 1 | + (−0.419 − 1.68i)3-s + (1.02 + 1.77i)5-s + (−2.64 + 0.105i)7-s + (−2.64 + 1.41i)9-s + (−5.11 − 2.95i)11-s + (0.139 + 0.0804i)13-s + (2.55 − 2.46i)15-s + (−2.77 − 4.81i)17-s + (−4.02 − 2.32i)19-s + (1.28 + 4.39i)21-s + (0.375 − 0.216i)23-s + (0.400 − 0.694i)25-s + (3.48 + 3.85i)27-s + (−1.95 + 1.12i)29-s + 2.97i·31-s + ⋯ |
L(s) = 1 | + (−0.242 − 0.970i)3-s + (0.458 + 0.793i)5-s + (−0.999 + 0.0398i)7-s + (−0.882 + 0.470i)9-s + (−1.54 − 0.890i)11-s + (0.0386 + 0.0223i)13-s + (0.658 − 0.636i)15-s + (−0.674 − 1.16i)17-s + (−0.922 − 0.532i)19-s + (0.280 + 0.959i)21-s + (0.0782 − 0.0451i)23-s + (0.0801 − 0.138i)25-s + (0.669 + 0.742i)27-s + (−0.363 + 0.209i)29-s + 0.534i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0534i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0534i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.00930430 - 0.347688i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00930430 - 0.347688i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.419 + 1.68i)T \) |
| 7 | \( 1 + (2.64 - 0.105i)T \) |
good | 5 | \( 1 + (-1.02 - 1.77i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (5.11 + 2.95i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.139 - 0.0804i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (2.77 + 4.81i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (4.02 + 2.32i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.375 + 0.216i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.95 - 1.12i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 2.97iT - 31T^{2} \) |
| 37 | \( 1 + (-2.17 + 3.77i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.35 - 4.08i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.82 - 3.15i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 0.130T + 47T^{2} \) |
| 53 | \( 1 + (10.7 - 6.23i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 6.44T + 59T^{2} \) |
| 61 | \( 1 + 6.90iT - 61T^{2} \) |
| 67 | \( 1 - 15.2T + 67T^{2} \) |
| 71 | \( 1 - 1.48iT - 71T^{2} \) |
| 73 | \( 1 + (2.60 - 1.50i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 - 17.3T + 79T^{2} \) |
| 83 | \( 1 + (7.62 + 13.2i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (4.04 - 7.01i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (2.61 - 1.50i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.83294737754423946691572821898, −9.639249513244584191705016573012, −8.604754026088129271036170310027, −7.58631728660133603916303300191, −6.67940025531089819527510457755, −6.11249968483627210164023387586, −5.03957269510272630197328383779, −3.02555157221128874022351796825, −2.46101202824975734801381721872, −0.19248082267395231801320911320,
2.29773486624907612023118815693, 3.74739601057663428986566269250, 4.76529534150695070056061234542, 5.63283060469113219180900088350, 6.51799760755870962729811252303, 7.979808711702374331900462918469, 8.882004829083388746454694958200, 9.739207247126405567108494453198, 10.30384681149052229729040817980, 11.04992026546777198781451806351