| L(s) = 1 | + (−389. + 224. i)2-s + (−6.71e3 + 9.16e3i)3-s + (3.54e4 − 6.13e4i)4-s + (6.52e5 + 1.12e6i)5-s + (5.54e5 − 5.07e6i)6-s + (1.13e6 + 1.52e7i)7-s − 2.70e7i·8-s + (−3.89e7 − 1.23e8i)9-s + (−5.07e8 − 2.92e8i)10-s + (7.67e8 + 4.43e8i)11-s + (3.24e8 + 7.36e8i)12-s + 1.92e9i·13-s + (−3.86e9 − 5.66e9i)14-s + (−1.47e10 − 1.60e9i)15-s + (1.07e10 + 1.85e10i)16-s + (−1.05e10 + 1.83e10i)17-s + ⋯ |
| L(s) = 1 | + (−1.07 + 0.620i)2-s + (−0.591 + 0.806i)3-s + (0.270 − 0.467i)4-s + (0.746 + 1.29i)5-s + (0.134 − 1.23i)6-s + (0.0747 + 0.997i)7-s − 0.570i·8-s + (−0.301 − 0.953i)9-s + (−1.60 − 0.926i)10-s + (1.07 + 0.623i)11-s + (0.217 + 0.494i)12-s + 0.653i·13-s + (−0.699 − 1.02i)14-s + (−1.48 − 0.162i)15-s + (0.624 + 1.08i)16-s + (−0.367 + 0.637i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0926 + 0.995i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (0.0926 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(0.7595443095\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7595443095\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (6.71e3 - 9.16e3i)T \) |
| 7 | \( 1 + (-1.13e6 - 1.52e7i)T \) |
| good | 2 | \( 1 + (389. - 224. i)T + (6.55e4 - 1.13e5i)T^{2} \) |
| 5 | \( 1 + (-6.52e5 - 1.12e6i)T + (-3.81e11 + 6.60e11i)T^{2} \) |
| 11 | \( 1 + (-7.67e8 - 4.43e8i)T + (2.52e17 + 4.37e17i)T^{2} \) |
| 13 | \( 1 - 1.92e9iT - 8.65e18T^{2} \) |
| 17 | \( 1 + (1.05e10 - 1.83e10i)T + (-4.13e20 - 7.16e20i)T^{2} \) |
| 19 | \( 1 + (1.02e11 - 5.93e10i)T + (2.74e21 - 4.74e21i)T^{2} \) |
| 23 | \( 1 + (2.29e11 - 1.32e11i)T + (7.05e22 - 1.22e23i)T^{2} \) |
| 29 | \( 1 - 2.01e12iT - 7.25e24T^{2} \) |
| 31 | \( 1 + (-1.38e12 - 7.99e11i)T + (1.12e25 + 1.95e25i)T^{2} \) |
| 37 | \( 1 + (6.92e12 + 1.19e13i)T + (-2.28e26 + 3.95e26i)T^{2} \) |
| 41 | \( 1 - 4.36e13T + 2.61e27T^{2} \) |
| 43 | \( 1 + 4.58e12T + 5.87e27T^{2} \) |
| 47 | \( 1 + (9.87e13 + 1.71e14i)T + (-1.33e28 + 2.30e28i)T^{2} \) |
| 53 | \( 1 + (-6.00e14 - 3.46e14i)T + (1.02e29 + 1.77e29i)T^{2} \) |
| 59 | \( 1 + (-1.85e14 + 3.21e14i)T + (-6.35e29 - 1.10e30i)T^{2} \) |
| 61 | \( 1 + (1.54e15 - 8.90e14i)T + (1.12e30 - 1.94e30i)T^{2} \) |
| 67 | \( 1 + (-1.02e15 + 1.76e15i)T + (-5.52e30 - 9.56e30i)T^{2} \) |
| 71 | \( 1 - 5.60e15iT - 2.96e31T^{2} \) |
| 73 | \( 1 + (-9.29e15 - 5.36e15i)T + (2.37e31 + 4.11e31i)T^{2} \) |
| 79 | \( 1 + (6.38e15 + 1.10e16i)T + (-9.09e31 + 1.57e32i)T^{2} \) |
| 83 | \( 1 - 3.82e16T + 4.21e32T^{2} \) |
| 89 | \( 1 + (-3.67e16 - 6.36e16i)T + (-6.89e32 + 1.19e33i)T^{2} \) |
| 97 | \( 1 + 7.45e16iT - 5.95e33T^{2} \) |
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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
−15.23506577871801557740116876778, −14.54482729743783137295114299655, −12.23365653196114838894636060719, −10.72810320885232847581357439720, −9.733590436513689645721901915550, −8.778514881812824285829852964582, −6.73250245850815798639312167042, −6.05773479606398236327301315424, −3.89100820202607032290236532250, −1.90434340722351658857077963479,
0.44494927004903053753905693619, 0.921022297231434398771283467317, 2.07072471702264489894218209926, 4.76214595221735092385552694121, 6.28489773819051555586006365743, 8.073669726664099352462172121805, 9.146972415605823820333998437107, 10.50354905779037299336444019863, 11.64337748386174819816825712207, 13.00878609223045732529946606613
