L(s) = 1 | + (1 − i)2-s − 2i·4-s + (2.20 + 4.48i)5-s + (−1.87 + 1.87i)7-s + (−2 − 2i)8-s + (6.69 + 2.28i)10-s − 15.1·11-s + (14.6 + 14.6i)13-s + 3.74i·14-s − 4·16-s + (−15.9 + 15.9i)17-s − 2.05i·19-s + (8.97 − 4.40i)20-s + (−15.1 + 15.1i)22-s + (−19.4 − 19.4i)23-s + ⋯ |
L(s) = 1 | + (0.5 − 0.5i)2-s − 0.5i·4-s + (0.440 + 0.897i)5-s + (−0.267 + 0.267i)7-s + (−0.250 − 0.250i)8-s + (0.669 + 0.228i)10-s − 1.37·11-s + (1.12 + 1.12i)13-s + 0.267i·14-s − 0.250·16-s + (−0.940 + 0.940i)17-s − 0.107i·19-s + (0.448 − 0.220i)20-s + (−0.687 + 0.687i)22-s + (−0.847 − 0.847i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0972 - 0.995i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0972 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.597589460\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.597589460\) |
\(L(2)\) |
|
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 | \( 1 \) |
| 5 | \( 1 + (-2.20 - 4.48i)T \) |
| 7 | \( 1 + (1.87 - 1.87i)T \) |
good | 11 | \( 1 + 15.1T + 121T^{2} \) |
| 13 | \( 1 + (-14.6 - 14.6i)T + 169iT^{2} \) |
| 17 | \( 1 + (15.9 - 15.9i)T - 289iT^{2} \) |
| 19 | \( 1 + 2.05iT - 361T^{2} \) |
| 23 | \( 1 + (19.4 + 19.4i)T + 529iT^{2} \) |
| 29 | \( 1 - 33.1iT - 841T^{2} \) |
| 31 | \( 1 - 27.8T + 961T^{2} \) |
| 37 | \( 1 + (30.4 - 30.4i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 - 26.6T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-7.89 - 7.89i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (33.8 - 33.8i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (5.60 + 5.60i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 - 5.80iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 98.2T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-51.6 + 51.6i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 - 120.T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-81.9 - 81.9i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 + 33.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-97.0 - 97.0i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + 34.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-104. + 104. i)T - 9.40e3iT^{2} \) |
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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
−10.76479444733984527039324706844, −10.01272456963590098309006663216, −8.982438573766659459414471651887, −8.035550132928799794692216579066, −6.57246951809009423813181920996, −6.26221409791183085311746676606, −5.02054569562281244196985919835, −3.86590508468933360358395373865, −2.77363868067898399726154403558, −1.83654819177214864732265808583,
0.46527423705379565782956124462, 2.34710964307425724665756660501, 3.64940818015524919471923901342, 4.82662306022568740468402750203, 5.57635888822726911910287925714, 6.34863755144194443163404381381, 7.70094422354478252293326197401, 8.207761171956189518063695363788, 9.232317320863832305571438231371, 10.15679935411308903524373887817