| L(s) = 1 | + (−5.89 − 44.8i)2-s + (137. − 137. i)3-s + (−1.97e3 + 529. i)4-s + (3.89e3 + 3.89e3i)5-s + (−6.95e3 − 5.34e3i)6-s + 3.79e4i·7-s + (3.54e4 + 8.56e4i)8-s + 1.39e5i·9-s + (1.51e5 − 1.97e5i)10-s + (−6.81e4 − 6.81e4i)11-s + (−1.98e5 + 3.43e5i)12-s + (−1.09e5 + 1.09e5i)13-s + (1.70e6 − 2.23e5i)14-s + 1.06e6·15-s + (3.63e6 − 2.09e6i)16-s + 8.85e6·17-s + ⋯ |
| L(s) = 1 | + (−0.130 − 0.991i)2-s + (0.325 − 0.325i)3-s + (−0.966 + 0.258i)4-s + (0.558 + 0.558i)5-s + (−0.365 − 0.280i)6-s + 0.853i·7-s + (0.382 + 0.924i)8-s + 0.787i·9-s + (0.480 − 0.626i)10-s + (−0.127 − 0.127i)11-s + (−0.230 + 0.398i)12-s + (−0.0820 + 0.0820i)13-s + (0.845 − 0.111i)14-s + 0.363·15-s + (0.866 − 0.499i)16-s + 1.51·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.129i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.991 - 0.129i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.67338 + 0.109000i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.67338 + 0.109000i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (5.89 + 44.8i)T \) |
| good | 3 | \( 1 + (-137. + 137. i)T - 1.77e5iT^{2} \) |
| 5 | \( 1 + (-3.89e3 - 3.89e3i)T + 4.88e7iT^{2} \) |
| 7 | \( 1 - 3.79e4iT - 1.97e9T^{2} \) |
| 11 | \( 1 + (6.81e4 + 6.81e4i)T + 2.85e11iT^{2} \) |
| 13 | \( 1 + (1.09e5 - 1.09e5i)T - 1.79e12iT^{2} \) |
| 17 | \( 1 - 8.85e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + (7.12e5 - 7.12e5i)T - 1.16e14iT^{2} \) |
| 23 | \( 1 - 2.69e7iT - 9.52e14T^{2} \) |
| 29 | \( 1 + (1.10e8 - 1.10e8i)T - 1.22e16iT^{2} \) |
| 31 | \( 1 + 1.15e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + (-3.15e8 - 3.15e8i)T + 1.77e17iT^{2} \) |
| 41 | \( 1 - 8.66e8iT - 5.50e17T^{2} \) |
| 43 | \( 1 + (3.13e8 + 3.13e8i)T + 9.29e17iT^{2} \) |
| 47 | \( 1 - 1.93e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + (-8.24e8 - 8.24e8i)T + 9.26e18iT^{2} \) |
| 59 | \( 1 + (7.47e9 + 7.47e9i)T + 3.01e19iT^{2} \) |
| 61 | \( 1 + (6.44e8 - 6.44e8i)T - 4.35e19iT^{2} \) |
| 67 | \( 1 + (-5.39e9 + 5.39e9i)T - 1.22e20iT^{2} \) |
| 71 | \( 1 + 2.90e10iT - 2.31e20T^{2} \) |
| 73 | \( 1 - 1.12e10iT - 3.13e20T^{2} \) |
| 79 | \( 1 + 2.03e7T + 7.47e20T^{2} \) |
| 83 | \( 1 + (-2.10e10 + 2.10e10i)T - 1.28e21iT^{2} \) |
| 89 | \( 1 - 7.01e10iT - 2.77e21T^{2} \) |
| 97 | \( 1 + 3.85e10T + 7.15e21T^{2} \) |
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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
−16.73749782631497655371729489423, −14.64166945115065800686553994946, −13.56480543371982971431335775501, −12.27128864725359897788301464271, −10.78882686668686610563133349056, −9.450688628755526302216500705427, −7.88617903190498291095874751514, −5.44086690756244471472456933560, −3.02173120898469245343548382973, −1.74171612001673448932990819012,
0.794601337149158505272984983294, 3.98114831089578837703192349779, 5.70530602659023159213176000139, 7.46022749603261001413403650379, 9.072679932167324688955588087513, 10.13178492563260375841808123919, 12.68900889733888213135763953025, 13.99567835720548556805382785558, 15.03873970101092484049340590028, 16.52358591687886153506047682267
