| L(s) = 1 | + (−5.58 + 4.05i)2-s + (−0.316 − 0.973i)3-s + (4.84 − 14.9i)4-s + (−20.2 − 14.6i)5-s + (5.71 + 4.15i)6-s + (−1.13 + 3.50i)7-s + (−34.8 − 107. i)8-s + (195. − 142. i)9-s + 172.·10-s + (−333. + 223. i)11-s − 16.0·12-s + (563. − 409. i)13-s + (−7.85 − 24.1i)14-s + (−7.90 + 24.3i)15-s + (1.03e3 + 752. i)16-s + (673. + 489. i)17-s + ⋯ |
| L(s) = 1 | + (−0.987 + 0.717i)2-s + (−0.0202 − 0.0624i)3-s + (0.151 − 0.466i)4-s + (−0.361 − 0.262i)5-s + (0.0648 + 0.0471i)6-s + (−0.00877 + 0.0269i)7-s + (−0.192 − 0.591i)8-s + (0.805 − 0.585i)9-s + 0.545·10-s + (−0.830 + 0.556i)11-s − 0.0321·12-s + (0.924 − 0.671i)13-s + (−0.0107 − 0.0329i)14-s + (−0.00907 + 0.0279i)15-s + (1.01 + 0.734i)16-s + (0.565 + 0.410i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.843 - 0.536i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.843 - 0.536i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.889447 + 0.259070i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.889447 + 0.259070i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (20.2 + 14.6i)T \) |
| 11 | \( 1 + (333. - 223. i)T \) |
| good | 2 | \( 1 + (5.58 - 4.05i)T + (9.88 - 30.4i)T^{2} \) |
| 3 | \( 1 + (0.316 + 0.973i)T + (-196. + 142. i)T^{2} \) |
| 7 | \( 1 + (1.13 - 3.50i)T + (-1.35e4 - 9.87e3i)T^{2} \) |
| 13 | \( 1 + (-563. + 409. i)T + (1.14e5 - 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-673. - 489. i)T + (4.38e5 + 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-835. - 2.57e3i)T + (-2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 - 3.03e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + (-2.43e3 + 7.50e3i)T + (-1.65e7 - 1.20e7i)T^{2} \) |
| 31 | \( 1 + (2.38e3 - 1.73e3i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (-4.83e3 + 1.48e4i)T + (-5.61e7 - 4.07e7i)T^{2} \) |
| 41 | \( 1 + (-3.14e3 - 9.67e3i)T + (-9.37e7 + 6.80e7i)T^{2} \) |
| 43 | \( 1 - 1.18e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (8.71e3 + 2.68e4i)T + (-1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (1.71e4 - 1.24e4i)T + (1.29e8 - 3.97e8i)T^{2} \) |
| 59 | \( 1 + (-1.14e3 + 3.51e3i)T + (-5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + (-4.12e4 - 2.99e4i)T + (2.60e8 + 8.03e8i)T^{2} \) |
| 67 | \( 1 - 2.07e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (2.97e4 + 2.15e4i)T + (5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + (-1.56e4 + 4.80e4i)T + (-1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (-1.12e4 + 8.14e3i)T + (9.50e8 - 2.92e9i)T^{2} \) |
| 83 | \( 1 + (-4.00e4 - 2.90e4i)T + (1.21e9 + 3.74e9i)T^{2} \) |
| 89 | \( 1 + 7.35e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (8.65e4 - 6.29e4i)T + (2.65e9 - 8.16e9i)T^{2} \) |
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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
−14.93058471275911462766305857747, −13.06171123975051157032998138228, −12.31023299841461463141814530078, −10.48796267540705118382289490526, −9.513892762633285987598044332499, −8.166653703164550392591070886657, −7.40604878669753518365946791003, −5.88964844997342157250885035981, −3.79011536853681773604582472092, −0.916945358094269284026411189743,
1.02597967108657834594387537851, 2.90750520014007527877607442993, 5.05203400137036950948621687286, 7.12580677289662493273278005622, 8.436294108543891580591174174283, 9.556984852476986555919407369939, 10.84102615817643284048959442504, 11.29162717508918293151617869438, 12.93420582355699698201874964731, 14.09379803471860203154130831491
