L(s) = 1 | + (2.28 − 2.28i)2-s + (−2.80 + 1.07i)3-s − 6.40i·4-s + (1.80 − 4.66i)5-s + (−3.93 + 8.83i)6-s + (−3.20 − 6.22i)7-s + (−5.48 − 5.48i)8-s + (6.69 − 6.01i)9-s + (−6.50 − 14.7i)10-s + 11.1i·11-s + (6.88 + 17.9i)12-s + (−5.82 − 5.82i)13-s + (−21.5 − 6.86i)14-s + (−0.0592 + 14.9i)15-s + 0.592·16-s + (6.84 + 6.84i)17-s + ⋯ |
L(s) = 1 | + (1.14 − 1.14i)2-s + (−0.933 + 0.358i)3-s − 1.60i·4-s + (0.361 − 0.932i)5-s + (−0.656 + 1.47i)6-s + (−0.458 − 0.888i)7-s + (−0.685 − 0.685i)8-s + (0.743 − 0.668i)9-s + (−0.650 − 1.47i)10-s + 1.01i·11-s + (0.573 + 1.49i)12-s + (−0.448 − 0.448i)13-s + (−1.53 − 0.490i)14-s + (−0.00395 + 0.999i)15-s + 0.0370·16-s + (0.402 + 0.402i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.518 + 0.855i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.518 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.892872 - 1.58527i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.892872 - 1.58527i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (2.80 - 1.07i)T \) |
| 5 | \( 1 + (-1.80 + 4.66i)T \) |
| 7 | \( 1 + (3.20 + 6.22i)T \) |
good | 2 | \( 1 + (-2.28 + 2.28i)T - 4iT^{2} \) |
| 11 | \( 1 - 11.1iT - 121T^{2} \) |
| 13 | \( 1 + (5.82 + 5.82i)T + 169iT^{2} \) |
| 17 | \( 1 + (-6.84 - 6.84i)T + 289iT^{2} \) |
| 19 | \( 1 - 25.0T + 361T^{2} \) |
| 23 | \( 1 + (-23.3 - 23.3i)T + 529iT^{2} \) |
| 29 | \( 1 + 10.6T + 841T^{2} \) |
| 31 | \( 1 - 26.9iT - 961T^{2} \) |
| 37 | \( 1 + (20.8 + 20.8i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 - 32.9T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-1.25 + 1.25i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (59.1 + 59.1i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (26.0 + 26.0i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 - 70.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 14.1iT - 3.72e3T^{2} \) |
| 67 | \( 1 + (6.14 + 6.14i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 - 39.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-51.1 - 51.1i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 - 16.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (31.3 - 31.3i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 - 70.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-114. + 114. i)T - 9.40e3iT^{2} \) |
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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
−12.83413089000031147749706164714, −12.33710537308209737054617657216, −11.29670027783871255120845644897, −10.15192238139616385647876259583, −9.623745271630045629072213015728, −7.25135352754120466003854675731, −5.53748296953441699360626929374, −4.81928401238143645654350010562, −3.60540697112723841483763225422, −1.25450594003756216512570658190,
3.08348372598808906320717482914, 5.04295222574483047241547681011, 5.97016142738505521181432636956, 6.66679091221384130521439151817, 7.72351018501950388059793211039, 9.593989020488294257871834722575, 11.09254316654064445467818509358, 12.04673386015024795102266974361, 13.06351787850773521761892283955, 13.93850753994870002147174417294