| L(s) = 1 | + (−6.81 + 6.32i)2-s + (−100. + 15.2i)3-s + (−31.7 + 424. i)4-s + (412. − 1.05e3i)5-s + (591. − 742. i)6-s + (5.12e3 + 3.76e3i)7-s + (−5.43e3 − 6.81e3i)8-s + (−8.85e3 + 2.73e3i)9-s + (3.84e3 + 9.78e3i)10-s + (−3.38e4 − 1.04e4i)11-s + (−3.24e3 − 4.33e4i)12-s + (−3.19e4 − 1.39e5i)13-s + (−5.86e4 + 6.75e3i)14-s + (−2.56e4 + 1.12e5i)15-s + (−1.35e5 − 2.03e4i)16-s + (1.61e5 − 1.09e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.301 + 0.279i)2-s + (−0.719 + 0.108i)3-s + (−0.0621 + 0.828i)4-s + (0.295 − 0.752i)5-s + (0.186 − 0.233i)6-s + (0.806 + 0.591i)7-s + (−0.469 − 0.588i)8-s + (−0.449 + 0.138i)9-s + (0.121 + 0.309i)10-s + (−0.697 − 0.215i)11-s + (−0.0451 − 0.602i)12-s + (−0.309 − 1.35i)13-s + (−0.408 + 0.0469i)14-s + (−0.130 + 0.573i)15-s + (−0.515 − 0.0777i)16-s + (0.468 − 0.319i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.121i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.992 - 0.121i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.11798 + 0.0684379i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.11798 + 0.0684379i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-5.12e3 - 3.76e3i)T \) |
| good | 2 | \( 1 + (6.81 - 6.32i)T + (38.2 - 510. i)T^{2} \) |
| 3 | \( 1 + (100. - 15.2i)T + (1.88e4 - 5.80e3i)T^{2} \) |
| 5 | \( 1 + (-412. + 1.05e3i)T + (-1.43e6 - 1.32e6i)T^{2} \) |
| 11 | \( 1 + (3.38e4 + 1.04e4i)T + (1.94e9 + 1.32e9i)T^{2} \) |
| 13 | \( 1 + (3.19e4 + 1.39e5i)T + (-9.55e9 + 4.60e9i)T^{2} \) |
| 17 | \( 1 + (-1.61e5 + 1.09e5i)T + (4.33e10 - 1.10e11i)T^{2} \) |
| 19 | \( 1 + (-2.50e5 - 4.33e5i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + (3.58e5 + 2.44e5i)T + (6.58e11 + 1.67e12i)T^{2} \) |
| 29 | \( 1 + (-1.30e6 - 6.27e5i)T + (9.04e12 + 1.13e13i)T^{2} \) |
| 31 | \( 1 + (-3.92e6 + 6.79e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 + (-1.57e6 - 2.09e7i)T + (-1.28e14 + 1.93e13i)T^{2} \) |
| 41 | \( 1 + (-1.27e7 - 1.60e7i)T + (-7.28e13 + 3.19e14i)T^{2} \) |
| 43 | \( 1 + (-2.10e6 + 2.63e6i)T + (-1.11e14 - 4.89e14i)T^{2} \) |
| 47 | \( 1 + (-3.93e7 + 3.64e7i)T + (8.36e13 - 1.11e15i)T^{2} \) |
| 53 | \( 1 + (-7.96e6 + 1.06e8i)T + (-3.26e15 - 4.91e14i)T^{2} \) |
| 59 | \( 1 + (5.78e6 + 1.47e7i)T + (-6.35e15 + 5.89e15i)T^{2} \) |
| 61 | \( 1 + (-1.16e7 - 1.54e8i)T + (-1.15e16 + 1.74e15i)T^{2} \) |
| 67 | \( 1 + (-9.17e7 + 1.58e8i)T + (-1.36e16 - 2.35e16i)T^{2} \) |
| 71 | \( 1 + (1.73e8 - 8.35e7i)T + (2.85e16 - 3.58e16i)T^{2} \) |
| 73 | \( 1 + (1.67e8 + 1.55e8i)T + (4.39e15 + 5.87e16i)T^{2} \) |
| 79 | \( 1 + (-1.67e8 - 2.90e8i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + (3.84e6 - 1.68e7i)T + (-1.68e17 - 8.11e16i)T^{2} \) |
| 89 | \( 1 + (-5.11e8 + 1.57e8i)T + (2.89e17 - 1.97e17i)T^{2} \) |
| 97 | \( 1 - 6.47e8T + 7.60e17T^{2} \) |
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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
−13.43746942296374021540366417827, −12.35628580331281094554944862474, −11.55487056453595379061401411754, −10.04790582532192165411988231197, −8.488543463206014142749634487365, −7.85522353077865651054981080516, −5.81196151984681879885533117603, −4.92332135623326539071040760511, −2.81453489946918653952722758460, −0.63315328445466819762537363220,
0.890437001065201265245920201866, 2.38132589136299558917849307296, 4.74828069581515638686336119416, 5.98014065494125921481565448592, 7.21804144361760955786927104814, 9.034807310215576336805413826537, 10.44390538741516251826320504063, 11.00373133354821612807922581808, 12.05719552030465164024129545384, 14.09976144797487673150230370828
