| L(s) = 1 | + (−2.35 + 3.23i)2-s + (−2.02 + 0.657i)3-s + (−4.94 − 15.2i)4-s + (24.5 − 50.2i)5-s + (2.63 − 8.09i)6-s + 244. i·7-s + (60.8 + 19.7i)8-s + (−192. + 140. i)9-s + (104. + 197. i)10-s + (−411. − 299. i)11-s + (20.0 + 27.5i)12-s + (−197. − 272. i)13-s + (−792. − 575. i)14-s + (−16.6 + 117. i)15-s + (−207. + 150. i)16-s + (−623. − 202. i)17-s + ⋯ |
| L(s) = 1 | + (−0.415 + 0.572i)2-s + (−0.129 + 0.0421i)3-s + (−0.154 − 0.475i)4-s + (0.439 − 0.898i)5-s + (0.0298 − 0.0918i)6-s + 1.88i·7-s + (0.336 + 0.109i)8-s + (−0.793 + 0.576i)9-s + (0.331 + 0.624i)10-s + (−1.02 − 0.745i)11-s + (0.0401 + 0.0552i)12-s + (−0.324 − 0.446i)13-s + (−1.08 − 0.784i)14-s + (−0.0191 + 0.135i)15-s + (−0.202 + 0.146i)16-s + (−0.523 − 0.170i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0923i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.0216433 + 0.467685i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0216433 + 0.467685i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.35 - 3.23i)T \) |
| 5 | \( 1 + (-24.5 + 50.2i)T \) |
| good | 3 | \( 1 + (2.02 - 0.657i)T + (196. - 142. i)T^{2} \) |
| 7 | \( 1 - 244. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + (411. + 299. i)T + (4.97e4 + 1.53e5i)T^{2} \) |
| 13 | \( 1 + (197. + 272. i)T + (-1.14e5 + 3.53e5i)T^{2} \) |
| 17 | \( 1 + (623. + 202. i)T + (1.14e6 + 8.34e5i)T^{2} \) |
| 19 | \( 1 + (721. - 2.22e3i)T + (-2.00e6 - 1.45e6i)T^{2} \) |
| 23 | \( 1 + (1.72e3 - 2.37e3i)T + (-1.98e6 - 6.12e6i)T^{2} \) |
| 29 | \( 1 + (105. + 325. i)T + (-1.65e7 + 1.20e7i)T^{2} \) |
| 31 | \( 1 + (1.37e3 - 4.21e3i)T + (-2.31e7 - 1.68e7i)T^{2} \) |
| 37 | \( 1 + (-5.81e3 - 8.00e3i)T + (-2.14e7 + 6.59e7i)T^{2} \) |
| 41 | \( 1 + (-1.70e3 + 1.23e3i)T + (3.58e7 - 1.10e8i)T^{2} \) |
| 43 | \( 1 + 2.19e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (-1.48e4 + 4.81e3i)T + (1.85e8 - 1.34e8i)T^{2} \) |
| 53 | \( 1 + (-1.37e4 + 4.46e3i)T + (3.38e8 - 2.45e8i)T^{2} \) |
| 59 | \( 1 + (-1.11e4 + 8.07e3i)T + (2.20e8 - 6.79e8i)T^{2} \) |
| 61 | \( 1 + (-5.61e3 - 4.07e3i)T + (2.60e8 + 8.03e8i)T^{2} \) |
| 67 | \( 1 + (-1.86e4 - 6.04e3i)T + (1.09e9 + 7.93e8i)T^{2} \) |
| 71 | \( 1 + (-2.48e4 - 7.66e4i)T + (-1.45e9 + 1.06e9i)T^{2} \) |
| 73 | \( 1 + (3.43e4 - 4.72e4i)T + (-6.40e8 - 1.97e9i)T^{2} \) |
| 79 | \( 1 + (2.83e3 + 8.73e3i)T + (-2.48e9 + 1.80e9i)T^{2} \) |
| 83 | \( 1 + (4.36e3 + 1.41e3i)T + (3.18e9 + 2.31e9i)T^{2} \) |
| 89 | \( 1 + (1.11e5 + 8.08e4i)T + (1.72e9 + 5.31e9i)T^{2} \) |
| 97 | \( 1 + (-7.80e3 + 2.53e3i)T + (6.94e9 - 5.04e9i)T^{2} \) |
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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
−15.43053284560594989040164150719, −14.06448160588005749650091562641, −12.82552249667345319494105137032, −11.68484337323784782388670971119, −10.10744567397610635341430358545, −8.715954883509693451820033788522, −8.206561086920842947865542554950, −5.69898967245302136790081998865, −5.44555791801775387478788498045, −2.29362353166343365988289891543,
0.25784726480704690332865777994, 2.53285531317710111904651561050, 4.28887104404099374156483021268, 6.61864383825655001341734545493, 7.65057239147817117656298415742, 9.497900791144028254142818011496, 10.57746474131124774861229076748, 11.18970101958044807056666322837, 12.90362951489554734975261263542, 13.86093446044542429924159656037
