| L(s) = 1 | + (4 − 6.92i)2-s + (6 + 10.3i)3-s + (−31.9 − 55.4i)4-s + (−105 + 181. i)5-s + 96·6-s − 511.·8-s + (1.02e3 − 1.76e3i)9-s + (840 + 1.45e3i)10-s + (−546 − 945. i)11-s + (384. − 665. i)12-s − 1.38e3·13-s − 2.52e3·15-s + (−2.04e3 + 3.54e3i)16-s + (7.35e3 + 1.27e4i)17-s + (−8.17e3 − 1.41e4i)18-s + (−1.99e4 + 3.45e4i)19-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (0.128 + 0.222i)3-s + (−0.249 − 0.433i)4-s + (−0.375 + 0.650i)5-s + 0.181·6-s − 0.353·8-s + (0.467 − 0.809i)9-s + (0.265 + 0.460i)10-s + (−0.123 − 0.214i)11-s + (0.0641 − 0.111i)12-s − 0.174·13-s − 0.192·15-s + (−0.125 + 0.216i)16-s + (0.362 + 0.628i)17-s + (−0.330 − 0.572i)18-s + (−0.667 + 1.15i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.13289 + 0.861852i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.13289 + 0.861852i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-4 + 6.92i)T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + (-6 - 10.3i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 + (105 - 181. i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 11 | \( 1 + (546 + 945. i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + 1.38e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + (-7.35e3 - 1.27e4i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (1.99e4 - 3.45e4i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (3.43e4 - 5.95e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + 1.02e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + (-1.13e5 - 1.97e5i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + (8.02e4 - 1.39e5i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + 1.08e4T + 1.94e11T^{2} \) |
| 43 | \( 1 + 6.30e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (-2.36e5 + 4.09e5i)T + (-2.53e11 - 4.38e11i)T^{2} \) |
| 53 | \( 1 + (-7.47e5 - 1.29e6i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 + (-1.32e6 - 2.28e6i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-4.13e5 + 7.16e5i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-6.30e4 - 1.09e5i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + 1.41e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + (-4.90e5 - 8.48e5i)T + (-5.52e12 + 9.56e12i)T^{2} \) |
| 79 | \( 1 + (-1.78e6 + 3.08e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 + 5.67e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (5.97e6 - 1.03e7i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + 8.68e6T + 8.07e13T^{2} \) |
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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
−12.55516311602722165105178996562, −11.76971977034836977793949136947, −10.57292256125683498915466844564, −9.832865226955374622410573937025, −8.456696712534841751743613664089, −7.02211500773927800202802300755, −5.72488250731941276301291371887, −4.04583766788921321359098881765, −3.21738116624079064952079907792, −1.48030893478338029826240809088,
0.39101113651558714819590160300, 2.36027130641956334447531448500, 4.26255192786789488338566611610, 5.12795365530475347596602945477, 6.71152000501577677689600751908, 7.78194782654030496789757840601, 8.667945695100362402960433352417, 10.03146377584521989716027422144, 11.46150716690675842475739605697, 12.63031718916774618443500625102
