L(s) = 1 | + (−0.309 + 0.951i)2-s + (−0.5 + 0.363i)3-s + (−0.809 − 0.587i)4-s + (−0.309 − 0.951i)5-s + (−0.190 − 0.587i)6-s + (−0.809 − 0.587i)7-s + (0.809 − 0.587i)8-s + (−0.809 + 2.48i)9-s + 0.999·10-s + (2.19 − 2.48i)11-s + 0.618·12-s + (−1.61 + 4.97i)13-s + (0.809 − 0.587i)14-s + (0.5 + 0.363i)15-s + (0.309 + 0.951i)16-s + (−0.5 − 1.53i)17-s + ⋯ |
L(s) = 1 | + (−0.218 + 0.672i)2-s + (−0.288 + 0.209i)3-s + (−0.404 − 0.293i)4-s + (−0.138 − 0.425i)5-s + (−0.0779 − 0.239i)6-s + (−0.305 − 0.222i)7-s + (0.286 − 0.207i)8-s + (−0.269 + 0.829i)9-s + 0.316·10-s + (0.660 − 0.750i)11-s + 0.178·12-s + (−0.448 + 1.38i)13-s + (0.216 − 0.157i)14-s + (0.129 + 0.0937i)15-s + (0.0772 + 0.237i)16-s + (−0.121 − 0.373i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.836 + 0.548i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.836 + 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0395976 - 0.132507i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0395976 - 0.132507i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.309 - 0.951i)T \) |
| 5 | \( 1 + (0.309 + 0.951i)T \) |
| 7 | \( 1 + (0.809 + 0.587i)T \) |
| 11 | \( 1 + (-2.19 + 2.48i)T \) |
good | 3 | \( 1 + (0.5 - 0.363i)T + (0.927 - 2.85i)T^{2} \) |
| 13 | \( 1 + (1.61 - 4.97i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (0.5 + 1.53i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-0.309 + 0.224i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + 8.47T + 23T^{2} \) |
| 29 | \( 1 + (3.23 + 2.35i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (1.38 - 4.25i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (7.85 + 5.70i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (6.97 - 5.06i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 4.61T + 43T^{2} \) |
| 47 | \( 1 + (-5.47 + 3.97i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (2 - 6.15i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-2.92 - 2.12i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (2.47 + 7.60i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + 6.09T + 67T^{2} \) |
| 71 | \( 1 + (-0.763 - 2.35i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (3.5 + 2.54i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-0.527 + 1.62i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-3.42 - 10.5i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 - 9.85T + 89T^{2} \) |
| 97 | \( 1 + (3.33 - 10.2i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−10.72298160152140929324525348312, −9.811827655061618183104177594845, −9.041667701780342938643960879210, −8.300389006033699837188105119767, −7.34170823022748628121132723929, −6.46988549593738776838516397097, −5.57635968605080073820213557594, −4.63666605358893546052627916550, −3.73852437391705222709004355582, −1.88541572452671039132428535357,
0.07499409271853391688244565101, 1.83519326927910516605314212812, 3.16883552945888590116645985384, 3.96663028541244997337203786977, 5.37681471450471154822027279526, 6.29927317452500345128251371497, 7.23679214184163141535298970761, 8.179863526692453802786019408695, 9.141390075859026593202239837390, 10.00711595044353647265070021315