L(s) = 1 | + (−1.17 + 0.790i)2-s + (1.37 − 1.37i)3-s + (0.750 − 1.85i)4-s + (−0.707 − 0.707i)5-s + (−0.523 + 2.68i)6-s − 2.73i·7-s + (0.584 + 2.76i)8-s − 0.755i·9-s + (1.38 + 0.270i)10-s + (4.12 + 4.12i)11-s + (−1.51 − 3.56i)12-s + (−1.37 + 1.37i)13-s + (2.16 + 3.20i)14-s − 1.93·15-s + (−2.87 − 2.78i)16-s − 4.94·17-s + ⋯ |
L(s) = 1 | + (−0.829 + 0.558i)2-s + (0.791 − 0.791i)3-s + (0.375 − 0.926i)4-s + (−0.316 − 0.316i)5-s + (−0.213 + 1.09i)6-s − 1.03i·7-s + (0.206 + 0.978i)8-s − 0.251i·9-s + (0.438 + 0.0855i)10-s + (1.24 + 1.24i)11-s + (−0.436 − 1.03i)12-s + (−0.382 + 0.382i)13-s + (0.577 + 0.857i)14-s − 0.500·15-s + (−0.718 − 0.695i)16-s − 1.20·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.367i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.929 + 0.367i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.792314 - 0.151082i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.792314 - 0.151082i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.17 - 0.790i)T \) |
| 5 | \( 1 + (0.707 + 0.707i)T \) |
good | 3 | \( 1 + (-1.37 + 1.37i)T - 3iT^{2} \) |
| 7 | \( 1 + 2.73iT - 7T^{2} \) |
| 11 | \( 1 + (-4.12 - 4.12i)T + 11iT^{2} \) |
| 13 | \( 1 + (1.37 - 1.37i)T - 13iT^{2} \) |
| 17 | \( 1 + 4.94T + 17T^{2} \) |
| 19 | \( 1 + (0.292 - 0.292i)T - 19iT^{2} \) |
| 23 | \( 1 + 1.64iT - 23T^{2} \) |
| 29 | \( 1 + (5.67 - 5.67i)T - 29iT^{2} \) |
| 31 | \( 1 - 3.95T + 31T^{2} \) |
| 37 | \( 1 + (-2.48 - 2.48i)T + 37iT^{2} \) |
| 41 | \( 1 + 8.40iT - 41T^{2} \) |
| 43 | \( 1 + (3.22 + 3.22i)T + 43iT^{2} \) |
| 47 | \( 1 + 5.19T + 47T^{2} \) |
| 53 | \( 1 + (-7.20 - 7.20i)T + 53iT^{2} \) |
| 59 | \( 1 + (6.41 + 6.41i)T + 59iT^{2} \) |
| 61 | \( 1 + (3.82 - 3.82i)T - 61iT^{2} \) |
| 67 | \( 1 + (-5.76 + 5.76i)T - 67iT^{2} \) |
| 71 | \( 1 + 7.92iT - 71T^{2} \) |
| 73 | \( 1 - 4.36iT - 73T^{2} \) |
| 79 | \( 1 + 5.56T + 79T^{2} \) |
| 83 | \( 1 + (0.516 - 0.516i)T - 83iT^{2} \) |
| 89 | \( 1 + 6.42iT - 89T^{2} \) |
| 97 | \( 1 + 9.44T + 97T^{2} \) |
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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
−14.39991178755739787337809902138, −13.54355396621173847692855270362, −12.23253159214325167159078213146, −10.84993919754072972700555144728, −9.509374994916072572143186310945, −8.566540071455203923156655452999, −7.29570378865105828726238381266, −6.84426869543393618934015711383, −4.47585443271381284196301224819, −1.76492294093804938746179044531,
2.75379488932940346509866388665, 3.96753835173743538524118515468, 6.39835206876916354259853669460, 8.190435306715580283801380366522, 8.977760513209140237208351944162, 9.732519834426879671420778363612, 11.19000455020586103077216171552, 11.86173398960946792684976253058, 13.34772620623547195091417190730, 14.76986539425492041855302247666