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
−9.732225552429595541674102868510, −9.076380987676144769567793393026, −7.86734425020647863909533904464, −7.45030983510006504576282594321, −6.24401915462440064462035944738, −5.48665265678561196870437543315, −4.75557001195935851138761064821, −3.21590721193818539286313688497, −2.54837704382073680732420390190, −0.919818538629635314271177550025,
1.34126042211011409169925412060, 2.45330581739027024958829081312, 3.63476668753347619839544312933, 5.02942672008161005189925025762, 5.39761017277139989837687416949, 6.48363971289059436791859566079, 7.51433952094441044896254623956, 8.112147853478171268529234621923, 9.401425379513398906899617936740, 9.633585045912876397311456802721