L(s) = 1 | + (0.875 + 0.875i)2-s + (−1.79 − 1.79i)3-s − 0.466i·4-s + (1.70 − 1.44i)5-s − 3.15i·6-s + (2.45 + 2.45i)7-s + (2.15 − 2.15i)8-s + 3.47i·9-s + (2.75 + 0.235i)10-s + (−0.839 + 0.839i)12-s + (0.522 − 0.522i)13-s + 4.29i·14-s + (−5.66 − 0.483i)15-s + 2.84·16-s + (−0.436 − 0.436i)17-s + (−3.04 + 3.04i)18-s + ⋯ |
L(s) = 1 | + (0.619 + 0.619i)2-s + (−1.03 − 1.03i)3-s − 0.233i·4-s + (0.764 − 0.644i)5-s − 1.28i·6-s + (0.926 + 0.926i)7-s + (0.763 − 0.763i)8-s + 1.15i·9-s + (0.872 + 0.0744i)10-s + (−0.242 + 0.242i)12-s + (0.145 − 0.145i)13-s + 1.14i·14-s + (−1.46 − 0.124i)15-s + 0.712·16-s + (−0.105 − 0.105i)17-s + (−0.716 + 0.716i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.504 + 0.863i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.504 + 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.59909 - 0.917967i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.59909 - 0.917967i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-1.70 + 1.44i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.875 - 0.875i)T + 2iT^{2} \) |
| 3 | \( 1 + (1.79 + 1.79i)T + 3iT^{2} \) |
| 7 | \( 1 + (-2.45 - 2.45i)T + 7iT^{2} \) |
| 13 | \( 1 + (-0.522 + 0.522i)T - 13iT^{2} \) |
| 17 | \( 1 + (0.436 + 0.436i)T + 17iT^{2} \) |
| 19 | \( 1 - 3.14T + 19T^{2} \) |
| 23 | \( 1 + (4.30 + 4.30i)T + 23iT^{2} \) |
| 29 | \( 1 + 2.90T + 29T^{2} \) |
| 31 | \( 1 - 3.03T + 31T^{2} \) |
| 37 | \( 1 + (-1.62 + 1.62i)T - 37iT^{2} \) |
| 41 | \( 1 + 1.02iT - 41T^{2} \) |
| 43 | \( 1 + (4.07 - 4.07i)T - 43iT^{2} \) |
| 47 | \( 1 + (-0.767 + 0.767i)T - 47iT^{2} \) |
| 53 | \( 1 + (-3.03 - 3.03i)T + 53iT^{2} \) |
| 59 | \( 1 - 7.40iT - 59T^{2} \) |
| 61 | \( 1 + 3.74iT - 61T^{2} \) |
| 67 | \( 1 + (9.39 - 9.39i)T - 67iT^{2} \) |
| 71 | \( 1 - 3.61T + 71T^{2} \) |
| 73 | \( 1 + (-0.380 + 0.380i)T - 73iT^{2} \) |
| 79 | \( 1 - 12.2T + 79T^{2} \) |
| 83 | \( 1 + (11.3 - 11.3i)T - 83iT^{2} \) |
| 89 | \( 1 + 14.6iT - 89T^{2} \) |
| 97 | \( 1 + (10.1 - 10.1i)T - 97iT^{2} \) |
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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
−10.66943710413677745551615832200, −9.669984349693164283062167345897, −8.548756766026044531570314448298, −7.58984905233015036416952175426, −6.54326322993783082271728660778, −5.82015662426589337968197017831, −5.37147467170619400503666401513, −4.51866986899543060602552448988, −2.11364937224770243390819090378, −1.07970312917037292470012816560,
1.77181344345825626250729468334, 3.36800642935685548075663938536, 4.24677451242681203382176567563, 5.05884747813724604300075867096, 5.86857512648287626377208277399, 7.13778732189242095402210708659, 8.075960626084601025070855170330, 9.560869218134917101326233145055, 10.27635874704688039127703958363, 10.95960614054321591619853466024