L(s) = 1 | + (−1 + i)2-s + (−1.22 − 1.22i)3-s − 2i·4-s + (−4.35 − 2.45i)5-s + 2.44·6-s + (−3.40 + 3.40i)7-s + (2 + 2i)8-s + 2.99i·9-s + (6.81 − 1.90i)10-s − 17.0·11-s + (−2.44 + 2.44i)12-s + (9.51 + 9.51i)13-s − 6.81i·14-s + (2.33 + 8.34i)15-s − 4·16-s + (−1.23 + 1.23i)17-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.5i)2-s + (−0.408 − 0.408i)3-s − 0.5i·4-s + (−0.871 − 0.490i)5-s + 0.408·6-s + (−0.486 + 0.486i)7-s + (0.250 + 0.250i)8-s + 0.333i·9-s + (0.681 − 0.190i)10-s − 1.55·11-s + (−0.204 + 0.204i)12-s + (0.732 + 0.732i)13-s − 0.486i·14-s + (0.155 + 0.556i)15-s − 0.250·16-s + (−0.0724 + 0.0724i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.277i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.960 + 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5781073824\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5781073824\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 - i)T \) |
| 3 | \( 1 + (1.22 + 1.22i)T \) |
| 5 | \( 1 + (4.35 + 2.45i)T \) |
| 23 | \( 1 + (3.39 + 3.39i)T \) |
good | 7 | \( 1 + (3.40 - 3.40i)T - 49iT^{2} \) |
| 11 | \( 1 + 17.0T + 121T^{2} \) |
| 13 | \( 1 + (-9.51 - 9.51i)T + 169iT^{2} \) |
| 17 | \( 1 + (1.23 - 1.23i)T - 289iT^{2} \) |
| 19 | \( 1 - 10.6iT - 361T^{2} \) |
| 29 | \( 1 + 16.4iT - 841T^{2} \) |
| 31 | \( 1 + 21.1T + 961T^{2} \) |
| 37 | \( 1 + (-0.589 + 0.589i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 - 44.7T + 1.68e3T^{2} \) |
| 43 | \( 1 + (49.7 + 49.7i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-59.0 + 59.0i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (38.1 + 38.1i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + 16.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 85.3T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-87.1 + 87.1i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 + 18.0T + 5.04e3T^{2} \) |
| 73 | \( 1 + (17.4 + 17.4i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 - 67.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-8.41 - 8.41i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 - 44.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-84.0 + 84.0i)T - 9.40e3iT^{2} \) |
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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
−10.23579349420142895066533649252, −9.152796568907248582232640317194, −8.341334619511213744212723585449, −7.72047135649954985026535682554, −6.79714025135540256450440701256, −5.80199064302433329593745581520, −5.03056519549730502141900888183, −3.73267384726656974773074423029, −2.13121944002756264287817039863, −0.46472936332166628577373353297,
0.63628976319314413333489020561, 2.75319958612297497714413829786, 3.53348060436159662535045594669, 4.61538160990410555838341643924, 5.80418038215087910681104436376, 7.02132978586775391694810860290, 7.75545401966969734573062501431, 8.563671248364239528184569803167, 9.706906128565053556038583187798, 10.55515148836357807592447288920