| L(s) = 1 | + 321. i·3-s + 1.71e4i·5-s + (7.17e4 − 9.32e4i)7-s + 4.27e5·9-s − 1.62e6·11-s + 4.55e6i·13-s − 5.52e6·15-s − 1.51e7i·17-s − 8.19e7i·19-s + (3.00e7 + 2.30e7i)21-s − 8.57e7·23-s − 4.99e7·25-s + 3.08e8i·27-s − 9.60e8·29-s + 1.52e9i·31-s + ⋯ |
| L(s) = 1 | + 0.441i·3-s + 1.09i·5-s + (0.609 − 0.792i)7-s + 0.805·9-s − 0.920·11-s + 0.944i·13-s − 0.484·15-s − 0.627i·17-s − 1.74i·19-s + (0.349 + 0.269i)21-s − 0.579·23-s − 0.204·25-s + 0.797i·27-s − 1.61·29-s + 1.72i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.609 + 0.792i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.609 + 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(0.1498464861\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1498464861\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-7.17e4 + 9.32e4i)T \) |
| good | 3 | \( 1 - 321. iT - 5.31e5T^{2} \) |
| 5 | \( 1 - 1.71e4iT - 2.44e8T^{2} \) |
| 11 | \( 1 + 1.62e6T + 3.13e12T^{2} \) |
| 13 | \( 1 - 4.55e6iT - 2.32e13T^{2} \) |
| 17 | \( 1 + 1.51e7iT - 5.82e14T^{2} \) |
| 19 | \( 1 + 8.19e7iT - 2.21e15T^{2} \) |
| 23 | \( 1 + 8.57e7T + 2.19e16T^{2} \) |
| 29 | \( 1 + 9.60e8T + 3.53e17T^{2} \) |
| 31 | \( 1 - 1.52e9iT - 7.87e17T^{2} \) |
| 37 | \( 1 + 1.82e9T + 6.58e18T^{2} \) |
| 41 | \( 1 + 6.08e8iT - 2.25e19T^{2} \) |
| 43 | \( 1 + 3.59e9T + 3.99e19T^{2} \) |
| 47 | \( 1 + 6.65e9iT - 1.16e20T^{2} \) |
| 53 | \( 1 - 1.15e10T + 4.91e20T^{2} \) |
| 59 | \( 1 - 5.91e10iT - 1.77e21T^{2} \) |
| 61 | \( 1 - 2.23e10iT - 2.65e21T^{2} \) |
| 67 | \( 1 + 5.96e10T + 8.18e21T^{2} \) |
| 71 | \( 1 + 6.47e10T + 1.64e22T^{2} \) |
| 73 | \( 1 - 1.47e10iT - 2.29e22T^{2} \) |
| 79 | \( 1 + 2.34e11T + 5.90e22T^{2} \) |
| 83 | \( 1 + 1.33e11iT - 1.06e23T^{2} \) |
| 89 | \( 1 + 3.77e11iT - 2.46e23T^{2} \) |
| 97 | \( 1 + 1.35e12iT - 6.93e23T^{2} \) |
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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
−10.77080270710696769819958408562, −10.15484763612880264223633260280, −8.907092439742245088456139724700, −7.25289439853534862775800722534, −7.00264423851672702739008814138, −5.13676694681826059558607412428, −4.17954702320531393512277094820, −2.94234895200392963100800386476, −1.65435166550491387622577431372, −0.03053892318284062944449669309,
1.31756033933233536141469896577, 2.13251943513356343793457156595, 3.90488539347295303046227166381, 5.19080596471788564300365825498, 5.93168597179605206406157614601, 7.83528887621070987758731331295, 8.127086328024664755003434170144, 9.508553551675793734481035287687, 10.56217695684441532082325077547, 11.96399857844043150522076448595
