L(s) = 1 | + (−1.41 + 0.0612i)2-s + (−1.42 − 1.42i)3-s + (1.99 − 0.172i)4-s + (0.391 − 0.391i)5-s + (2.10 + 1.92i)6-s − 3.36·7-s + (−2.80 + 0.366i)8-s + 1.07i·9-s + (−0.528 + 0.576i)10-s + (−0.0339 − 0.0339i)11-s + (−3.09 − 2.59i)12-s + (−0.227 + 0.227i)13-s + (4.74 − 0.205i)14-s − 1.11·15-s + (3.94 − 0.689i)16-s + 0.208·17-s + ⋯ |
L(s) = 1 | + (−0.999 + 0.0432i)2-s + (−0.824 − 0.824i)3-s + (0.996 − 0.0864i)4-s + (0.174 − 0.174i)5-s + (0.859 + 0.787i)6-s − 1.27·7-s + (−0.991 + 0.129i)8-s + 0.359i·9-s + (−0.167 + 0.182i)10-s + (−0.0102 − 0.0102i)11-s + (−0.892 − 0.749i)12-s + (−0.0631 + 0.0631i)13-s + (1.26 − 0.0549i)14-s − 0.288·15-s + (0.985 − 0.172i)16-s + 0.0505·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.619 - 0.785i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.619 - 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0159665 + 0.0329218i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0159665 + 0.0329218i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.41 - 0.0612i)T \) |
| 19 | \( 1 + (-0.442 - 4.33i)T \) |
good | 3 | \( 1 + (1.42 + 1.42i)T + 3iT^{2} \) |
| 5 | \( 1 + (-0.391 + 0.391i)T - 5iT^{2} \) |
| 7 | \( 1 + 3.36T + 7T^{2} \) |
| 11 | \( 1 + (0.0339 + 0.0339i)T + 11iT^{2} \) |
| 13 | \( 1 + (0.227 - 0.227i)T - 13iT^{2} \) |
| 17 | \( 1 - 0.208T + 17T^{2} \) |
| 23 | \( 1 + 8.50T + 23T^{2} \) |
| 29 | \( 1 + (2.09 - 2.09i)T - 29iT^{2} \) |
| 31 | \( 1 + 0.695T + 31T^{2} \) |
| 37 | \( 1 + (-3.52 - 3.52i)T + 37iT^{2} \) |
| 41 | \( 1 + 7.19T + 41T^{2} \) |
| 43 | \( 1 + (-5.14 - 5.14i)T + 43iT^{2} \) |
| 47 | \( 1 + 6.18iT - 47T^{2} \) |
| 53 | \( 1 + (2.19 + 2.19i)T + 53iT^{2} \) |
| 59 | \( 1 + (-1.98 + 1.98i)T - 59iT^{2} \) |
| 61 | \( 1 + (5.48 + 5.48i)T + 61iT^{2} \) |
| 67 | \( 1 + (-0.280 - 0.280i)T + 67iT^{2} \) |
| 71 | \( 1 + 12.8iT - 71T^{2} \) |
| 73 | \( 1 + 7.46iT - 73T^{2} \) |
| 79 | \( 1 + 8.31T + 79T^{2} \) |
| 83 | \( 1 + (9.37 - 9.37i)T - 83iT^{2} \) |
| 89 | \( 1 + 7.02T + 89T^{2} \) |
| 97 | \( 1 - 5.52iT - 97T^{2} \) |
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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
−12.08562080947190207824352357040, −11.12153484142226907107403080069, −9.996331435311491785877259118908, −9.446048601489759632290833791452, −8.152220914482330643826712706996, −7.15957983461926862221192804027, −6.30209959803855294407310323154, −5.70183037565604594657018829766, −3.42234202353105717554779170340, −1.66942694228332526263347365005,
0.03856615433309215424999960981, 2.57872636154724321090748589422, 4.06054313961240100417892905041, 5.71287767217192963113279588291, 6.39553217380107729763518095316, 7.52862332673818263946190078734, 8.832625690301724829169263538883, 9.892352992836428572659305878976, 10.13364854602792178518598649235, 11.14887451056759958898897177281