| L(s) = 1 | + (−13.0 − 23.6i)3-s + (49.0 − 28.3i)5-s + (−226. + 392. i)7-s + (−388. + 617. i)9-s + (1.53e3 + 889. i)11-s + (−1.44e3 − 2.50e3i)13-s + (−1.31e3 − 789. i)15-s + 3.83e3i·17-s + 1.16e4·19-s + (1.22e4 + 230. i)21-s + (1.05e4 − 6.06e3i)23-s + (−6.20e3 + 1.07e4i)25-s + (1.96e4 + 1.11e3i)27-s + (−821. − 474. i)29-s + (1.56e4 + 2.70e4i)31-s + ⋯ |
| L(s) = 1 | + (−0.483 − 0.875i)3-s + (0.392 − 0.226i)5-s + (−0.660 + 1.14i)7-s + (−0.532 + 0.846i)9-s + (1.15 + 0.667i)11-s + (−0.658 − 1.14i)13-s + (−0.388 − 0.234i)15-s + 0.780i·17-s + 1.69·19-s + (1.32 + 0.0249i)21-s + (0.863 − 0.498i)23-s + (−0.397 + 0.688i)25-s + (0.998 + 0.0566i)27-s + (−0.0337 − 0.0194i)29-s + (0.524 + 0.908i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 - 0.306i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.951 - 0.306i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.46702 + 0.230195i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.46702 + 0.230195i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (13.0 + 23.6i)T \) |
| good | 5 | \( 1 + (-49.0 + 28.3i)T + (7.81e3 - 1.35e4i)T^{2} \) |
| 7 | \( 1 + (226. - 392. i)T + (-5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (-1.53e3 - 889. i)T + (8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 + (1.44e3 + 2.50e3i)T + (-2.41e6 + 4.18e6i)T^{2} \) |
| 17 | \( 1 - 3.83e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 1.16e4T + 4.70e7T^{2} \) |
| 23 | \( 1 + (-1.05e4 + 6.06e3i)T + (7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 + (821. + 474. i)T + (2.97e8 + 5.15e8i)T^{2} \) |
| 31 | \( 1 + (-1.56e4 - 2.70e4i)T + (-4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 - 7.24e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + (4.89e4 - 2.82e4i)T + (2.37e9 - 4.11e9i)T^{2} \) |
| 43 | \( 1 + (-1.43e4 + 2.48e4i)T + (-3.16e9 - 5.47e9i)T^{2} \) |
| 47 | \( 1 + (-6.26e4 - 3.61e4i)T + (5.38e9 + 9.33e9i)T^{2} \) |
| 53 | \( 1 + 1.06e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + (2.89e5 - 1.67e5i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (5.03e4 - 8.72e4i)T + (-2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-1.89e5 - 3.27e5i)T + (-4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 - 3.44e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 1.12e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + (-7.53e4 + 1.30e5i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + (8.10e5 + 4.67e5i)T + (1.63e11 + 2.83e11i)T^{2} \) |
| 89 | \( 1 - 6.56e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + (-3.31e5 + 5.74e5i)T + (-4.16e11 - 7.21e11i)T^{2} \) |
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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
−13.12923253373434872936938372841, −12.42427673760272253434420215061, −11.63524328534793328084971663433, −9.992353587600265527877421609627, −8.911241042820979202518550088000, −7.43763618517600803647823978451, −6.19268305930897642886304831988, −5.22881696938179766996654224872, −2.81414394273090561769608564688, −1.22417229329888812968954988627,
0.72939140293766032593813611106, 3.31285983141034863155444476605, 4.52706074650304115299763977373, 6.13163683710355631877173095261, 7.18605744665605224232220981488, 9.371169888107544289740660795718, 9.754779499372972534415514494187, 11.15910208127393116940663969955, 11.91888868069009048896133608903, 13.74648219917540323072508263276
