L(s) = 1 | + (0.894 + 1.09i)2-s + (0.860 + 1.49i)3-s + (−0.399 + 1.95i)4-s + (0.637 + 0.368i)5-s + (−0.862 + 2.27i)6-s + (2.61 − 0.368i)7-s + (−2.50 + 1.31i)8-s + (0.0185 − 0.0321i)9-s + (0.167 + 1.02i)10-s + (−0.866 + 0.5i)11-s + (−3.26 + 1.09i)12-s − 6.51i·13-s + (2.74 + 2.53i)14-s + 1.26i·15-s + (−3.68 − 1.56i)16-s + (−6.17 + 3.56i)17-s + ⋯ |
L(s) = 1 | + (0.632 + 0.774i)2-s + (0.496 + 0.860i)3-s + (−0.199 + 0.979i)4-s + (0.285 + 0.164i)5-s + (−0.352 + 0.929i)6-s + (0.990 − 0.139i)7-s + (−0.885 + 0.465i)8-s + (0.00618 − 0.0107i)9-s + (0.0529 + 0.325i)10-s + (−0.261 + 0.150i)11-s + (−0.942 + 0.315i)12-s − 1.80i·13-s + (0.734 + 0.678i)14-s + 0.327i·15-s + (−0.920 − 0.391i)16-s + (−1.49 + 0.864i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.329 - 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.329 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.25664 + 1.76951i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.25664 + 1.76951i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.894 - 1.09i)T \) |
| 7 | \( 1 + (-2.61 + 0.368i)T \) |
| 11 | \( 1 + (0.866 - 0.5i)T \) |
good | 3 | \( 1 + (-0.860 - 1.49i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.637 - 0.368i)T + (2.5 + 4.33i)T^{2} \) |
| 13 | \( 1 + 6.51iT - 13T^{2} \) |
| 17 | \( 1 + (6.17 - 3.56i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.897 + 1.55i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.642 - 0.370i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 2.50T + 29T^{2} \) |
| 31 | \( 1 + (-1.80 - 3.13i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.85 - 3.20i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 4.59iT - 41T^{2} \) |
| 43 | \( 1 - 5.47iT - 43T^{2} \) |
| 47 | \( 1 + (-4.59 + 7.96i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.36 - 9.28i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.626 - 1.08i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.79 + 2.18i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (13.0 - 7.53i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 14.7iT - 71T^{2} \) |
| 73 | \( 1 + (-11.9 + 6.91i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (3.37 + 1.94i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 2.45T + 83T^{2} \) |
| 89 | \( 1 + (6.91 + 3.99i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 11.6iT - 97T^{2} \) |
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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
−12.17306094288079834274910461293, −10.89846072573616438490867701454, −10.17711583441134569866502928136, −8.833707553352336822612702303698, −8.246122445673596317468934962407, −7.16800787097521238341880126307, −5.87927612472088352922790503567, −4.86121371672673567471296220320, −3.97104974584437462585450299915, −2.68903727393193695265931099504,
1.66723688254743543821933616574, 2.37640851618760655237674535231, 4.21177677104705527039942550544, 5.12424755652369492808308167042, 6.49106799406704338699311024105, 7.47901151322147309711768890027, 8.783788897384393173947601151658, 9.433643077916801285311973434851, 10.88335408084989319654883719250, 11.50875551434478744510764464564