L(s) = 1 | + (1.65 − 0.686i)5-s + (0.456 + 0.456i)7-s + (−2.79 + 1.15i)11-s + (−1.29 + 3.12i)13-s + 3.55·17-s + (5.61 + 2.32i)19-s + (4.10 + 4.10i)23-s + (−1.26 + 1.26i)25-s + (1.87 − 4.53i)29-s + 0.580i·31-s + (1.06 + 0.442i)35-s + (2.58 + 6.22i)37-s + (2.98 − 2.98i)41-s + (2.78 + 6.72i)43-s − 8.67i·47-s + ⋯ |
L(s) = 1 | + (0.740 − 0.306i)5-s + (0.172 + 0.172i)7-s + (−0.841 + 0.348i)11-s + (−0.359 + 0.867i)13-s + 0.862·17-s + (1.28 + 0.533i)19-s + (0.855 + 0.855i)23-s + (−0.252 + 0.252i)25-s + (0.348 − 0.841i)29-s + 0.104i·31-s + (0.180 + 0.0748i)35-s + (0.424 + 1.02i)37-s + (0.465 − 0.465i)41-s + (0.424 + 1.02i)43-s − 1.26i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.861 - 0.507i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.861 - 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.851869619\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.851869619\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-1.65 + 0.686i)T + (3.53 - 3.53i)T^{2} \) |
| 7 | \( 1 + (-0.456 - 0.456i)T + 7iT^{2} \) |
| 11 | \( 1 + (2.79 - 1.15i)T + (7.77 - 7.77i)T^{2} \) |
| 13 | \( 1 + (1.29 - 3.12i)T + (-9.19 - 9.19i)T^{2} \) |
| 17 | \( 1 - 3.55T + 17T^{2} \) |
| 19 | \( 1 + (-5.61 - 2.32i)T + (13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (-4.10 - 4.10i)T + 23iT^{2} \) |
| 29 | \( 1 + (-1.87 + 4.53i)T + (-20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 - 0.580iT - 31T^{2} \) |
| 37 | \( 1 + (-2.58 - 6.22i)T + (-26.1 + 26.1i)T^{2} \) |
| 41 | \( 1 + (-2.98 + 2.98i)T - 41iT^{2} \) |
| 43 | \( 1 + (-2.78 - 6.72i)T + (-30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + 8.67iT - 47T^{2} \) |
| 53 | \( 1 + (3.70 + 8.95i)T + (-37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (-1.77 - 4.28i)T + (-41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (9.49 + 3.93i)T + (43.1 + 43.1i)T^{2} \) |
| 67 | \( 1 + (3.50 - 8.46i)T + (-47.3 - 47.3i)T^{2} \) |
| 71 | \( 1 + (-7.84 + 7.84i)T - 71iT^{2} \) |
| 73 | \( 1 + (-10.7 - 10.7i)T + 73iT^{2} \) |
| 79 | \( 1 - 5.27T + 79T^{2} \) |
| 83 | \( 1 + (-1.80 + 4.36i)T + (-58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (-12.4 - 12.4i)T + 89iT^{2} \) |
| 97 | \( 1 - 9.99T + 97T^{2} \) |
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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
−9.633585045912876397311456802721, −9.401425379513398906899617936740, −8.112147853478171268529234621923, −7.51433952094441044896254623956, −6.48363971289059436791859566079, −5.39761017277139989837687416949, −5.02942672008161005189925025762, −3.63476668753347619839544312933, −2.45330581739027024958829081312, −1.34126042211011409169925412060,
0.919818538629635314271177550025, 2.54837704382073680732420390190, 3.21590721193818539286313688497, 4.75557001195935851138761064821, 5.48665265678561196870437543315, 6.24401915462440064462035944738, 7.45030983510006504576282594321, 7.86734425020647863909533904464, 9.076380987676144769567793393026, 9.732225552429595541674102868510