| L(s) = 1 | + (−2.71 − 11.8i)2-s + (111. + 103. i)3-s + (327. − 157. i)4-s + (−464. + 70.0i)5-s + (929. − 1.61e3i)6-s + (−1.14e3 − 1.98e3i)7-s + (−6.65e3 − 8.34e3i)8-s + (268. + 3.57e3i)9-s + (2.09e3 + 5.33e3i)10-s + (−7.80e4 − 3.76e4i)11-s + (5.29e4 + 1.63e4i)12-s + (2.30e4 − 5.88e4i)13-s + (−2.04e4 + 1.89e4i)14-s + (−5.92e4 − 4.04e4i)15-s + (3.49e4 − 4.38e4i)16-s + (3.01e5 + 4.54e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.119 − 0.525i)2-s + (0.797 + 0.739i)3-s + (0.639 − 0.307i)4-s + (−0.332 + 0.0501i)5-s + (0.292 − 0.507i)6-s + (−0.180 − 0.311i)7-s + (−0.574 − 0.720i)8-s + (0.0136 + 0.181i)9-s + (0.0662 + 0.168i)10-s + (−1.60 − 0.774i)11-s + (0.737 + 0.227i)12-s + (0.224 − 0.571i)13-s + (−0.142 + 0.131i)14-s + (−0.302 − 0.206i)15-s + (0.133 − 0.167i)16-s + (0.875 + 0.131i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.400 + 0.916i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.400 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.05044 - 1.60613i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05044 - 1.60613i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-2.14e7 - 6.48e6i)T \) |
| good | 2 | \( 1 + (2.71 + 11.8i)T + (-461. + 222. i)T^{2} \) |
| 3 | \( 1 + (-111. - 103. i)T + (1.47e3 + 1.96e4i)T^{2} \) |
| 5 | \( 1 + (464. - 70.0i)T + (1.86e6 - 5.75e5i)T^{2} \) |
| 7 | \( 1 + (1.14e3 + 1.98e3i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + (7.80e4 + 3.76e4i)T + (1.47e9 + 1.84e9i)T^{2} \) |
| 13 | \( 1 + (-2.30e4 + 5.88e4i)T + (-7.77e9 - 7.21e9i)T^{2} \) |
| 17 | \( 1 + (-3.01e5 - 4.54e4i)T + (1.13e11 + 3.49e10i)T^{2} \) |
| 19 | \( 1 + (271. - 3.62e3i)T + (-3.19e11 - 4.80e10i)T^{2} \) |
| 23 | \( 1 + (-1.93e6 + 1.31e6i)T + (6.58e11 - 1.67e12i)T^{2} \) |
| 29 | \( 1 + (-7.00e5 + 6.50e5i)T + (1.08e12 - 1.44e13i)T^{2} \) |
| 31 | \( 1 + (7.76e6 + 2.39e6i)T + (2.18e13 + 1.48e13i)T^{2} \) |
| 37 | \( 1 + (1.01e7 - 1.75e7i)T + (-6.49e13 - 1.12e14i)T^{2} \) |
| 41 | \( 1 + (6.78e6 + 2.97e7i)T + (-2.94e14 + 1.42e14i)T^{2} \) |
| 47 | \( 1 + (-4.76e7 + 2.29e7i)T + (6.97e14 - 8.74e14i)T^{2} \) |
| 53 | \( 1 + (-3.42e7 - 8.71e7i)T + (-2.41e15 + 2.24e15i)T^{2} \) |
| 59 | \( 1 + (-1.24e7 + 1.55e7i)T + (-1.92e15 - 8.44e15i)T^{2} \) |
| 61 | \( 1 + (1.25e8 - 3.88e7i)T + (9.66e15 - 6.58e15i)T^{2} \) |
| 67 | \( 1 + (2.20e6 - 2.94e7i)T + (-2.69e16 - 4.05e15i)T^{2} \) |
| 71 | \( 1 + (2.01e8 + 1.37e8i)T + (1.67e16 + 4.26e16i)T^{2} \) |
| 73 | \( 1 + (3.92e7 - 9.99e7i)T + (-4.31e16 - 4.00e16i)T^{2} \) |
| 79 | \( 1 + (-8.23e7 - 1.42e8i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + (-2.10e8 - 1.94e8i)T + (1.39e16 + 1.86e17i)T^{2} \) |
| 89 | \( 1 + (-3.26e8 - 3.03e8i)T + (2.61e16 + 3.49e17i)T^{2} \) |
| 97 | \( 1 + (9.23e7 + 4.44e7i)T + (4.73e17 + 5.94e17i)T^{2} \) |
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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
−13.63576061383142854133602890755, −12.32535865918970348434810630213, −10.77613333168588528692911186424, −10.25440194599146540076420180668, −8.816277350481733686316769345324, −7.45991810856113656172888675077, −5.60325235798712283211549410068, −3.56815938418264295339859322187, −2.69442995103770135417584835566, −0.58591946727312777350505215929,
1.93531572655787024189191584270, 3.07220466655517500346034581686, 5.45431726222714364191301473239, 7.28185878862493971798882130432, 7.69390487266178091540340046229, 9.002194425058905716396473017281, 10.82651845157691555644753877052, 12.25440153786032075438899502179, 13.13563777964925439727840370445, 14.48143739261342308451183878484
