L(s) = 1 | + (−8.20 + 4.73i)5-s + (1.05 − 1.83i)7-s + (13.7 + 7.91i)11-s + (−4.70 − 8.14i)13-s − 11.6i·17-s − 12.9·19-s + (5.27 − 3.04i)23-s + (32.4 − 56.1i)25-s + (−24.7 − 14.2i)29-s + (8.75 + 15.1i)31-s + 20.0i·35-s + 15.6·37-s + (−14.8 + 8.54i)41-s + (21.7 − 37.6i)43-s + (−20.6 − 11.9i)47-s + ⋯ |
L(s) = 1 | + (−1.64 + 0.947i)5-s + (0.150 − 0.261i)7-s + (1.24 + 0.719i)11-s + (−0.361 − 0.626i)13-s − 0.682i·17-s − 0.682·19-s + (0.229 − 0.132i)23-s + (1.29 − 2.24i)25-s + (−0.854 − 0.493i)29-s + (0.282 + 0.489i)31-s + 0.572i·35-s + 0.422·37-s + (−0.361 + 0.208i)41-s + (0.505 − 0.874i)43-s + (−0.439 − 0.253i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.743 - 0.668i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.743 - 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.249473178\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.249473178\) |
\(L(2)\) |
|
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 | \( 1 + (8.20 - 4.73i)T + (12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (-1.05 + 1.83i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-13.7 - 7.91i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (4.70 + 8.14i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + 11.6iT - 289T^{2} \) |
| 19 | \( 1 + 12.9T + 361T^{2} \) |
| 23 | \( 1 + (-5.27 + 3.04i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (24.7 + 14.2i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-8.75 - 15.1i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 15.6T + 1.36e3T^{2} \) |
| 41 | \( 1 + (14.8 - 8.54i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-21.7 + 37.6i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (20.6 + 11.9i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 - 14.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-38.5 + 22.2i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-1.86 + 3.22i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (21.0 + 36.3i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 120. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 5.48T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-60.5 + 104. i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-46.5 - 26.8i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 - 102. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (58.9 - 102. i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−9.181736984122968586803077664803, −8.272608970980287606482364539814, −7.49998906423324265533665079640, −7.04736148595048774947893638186, −6.29270416942635528868504642066, −4.86571177282727794444385499716, −4.09899686989767632262998277053, −3.45045265044303906298965590567, −2.36624881950427896347338328540, −0.69834515623225336019608680188,
0.55290336676679586601001058861, 1.69540286260085326358642933717, 3.36117843499160567250211420311, 4.07881737448500816307567111665, 4.66434682912769943033815500486, 5.77894739633537050003130532587, 6.75280446418942999076328480002, 7.58464166919580164177615122467, 8.407879633405374805006484406773, 8.828174205004521295555184909061