| L(s) = 1 | + (−6.46 + 17.7i)3-s + (14.8 + 84.2i)5-s + (158. − 273. i)7-s + (285. + 239. i)9-s + (1.05e3 + 1.83e3i)11-s + (−564. − 1.55e3i)13-s + (−1.59e3 − 280. i)15-s + (−3.14e3 + 2.64e3i)17-s + (−6.56e3 − 1.98e3i)19-s + (3.83e3 + 4.57e3i)21-s + (−3.59e3 + 2.04e4i)23-s + (7.81e3 − 2.84e3i)25-s + (−1.80e4 + 1.04e4i)27-s + (−8.79e3 + 1.04e4i)29-s + (−1.87e4 − 1.08e4i)31-s + ⋯ |
| L(s) = 1 | + (−0.239 + 0.657i)3-s + (0.118 + 0.673i)5-s + (0.460 − 0.798i)7-s + (0.391 + 0.328i)9-s + (0.794 + 1.37i)11-s + (−0.257 − 0.706i)13-s + (−0.471 − 0.0831i)15-s + (−0.640 + 0.537i)17-s + (−0.957 − 0.289i)19-s + (0.414 + 0.494i)21-s + (−0.295 + 1.67i)23-s + (0.499 − 0.181i)25-s + (−0.915 + 0.528i)27-s + (−0.360 + 0.429i)29-s + (−0.628 − 0.362i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.546 - 0.837i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.546 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.737306 + 1.36191i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.737306 + 1.36191i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (6.56e3 + 1.98e3i)T \) |
| good | 3 | \( 1 + (6.46 - 17.7i)T + (-558. - 468. i)T^{2} \) |
| 5 | \( 1 + (-14.8 - 84.2i)T + (-1.46e4 + 5.34e3i)T^{2} \) |
| 7 | \( 1 + (-158. + 273. i)T + (-5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (-1.05e3 - 1.83e3i)T + (-8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 + (564. + 1.55e3i)T + (-3.69e6 + 3.10e6i)T^{2} \) |
| 17 | \( 1 + (3.14e3 - 2.64e3i)T + (4.19e6 - 2.37e7i)T^{2} \) |
| 23 | \( 1 + (3.59e3 - 2.04e4i)T + (-1.39e8 - 5.06e7i)T^{2} \) |
| 29 | \( 1 + (8.79e3 - 1.04e4i)T + (-1.03e8 - 5.85e8i)T^{2} \) |
| 31 | \( 1 + (1.87e4 + 1.08e4i)T + (4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 - 8.06e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + (1.93e4 - 5.31e4i)T + (-3.63e9 - 3.05e9i)T^{2} \) |
| 43 | \( 1 + (2.21e4 + 1.25e5i)T + (-5.94e9 + 2.16e9i)T^{2} \) |
| 47 | \( 1 + (-1.15e5 - 9.66e4i)T + (1.87e9 + 1.06e10i)T^{2} \) |
| 53 | \( 1 + (-3.72e4 - 6.57e3i)T + (2.08e10 + 7.58e9i)T^{2} \) |
| 59 | \( 1 + (2.56e5 + 3.06e5i)T + (-7.32e9 + 4.15e10i)T^{2} \) |
| 61 | \( 1 + (-1.11e4 + 6.32e4i)T + (-4.84e10 - 1.76e10i)T^{2} \) |
| 67 | \( 1 + (-1.45e5 + 1.73e5i)T + (-1.57e10 - 8.90e10i)T^{2} \) |
| 71 | \( 1 + (-2.57e5 + 4.53e4i)T + (1.20e11 - 4.38e10i)T^{2} \) |
| 73 | \( 1 + (-5.14e5 - 1.87e5i)T + (1.15e11 + 9.72e10i)T^{2} \) |
| 79 | \( 1 + (-1.85e5 + 5.11e5i)T + (-1.86e11 - 1.56e11i)T^{2} \) |
| 83 | \( 1 + (2.44e5 - 4.23e5i)T + (-1.63e11 - 2.83e11i)T^{2} \) |
| 89 | \( 1 + (-3.10e4 - 8.52e4i)T + (-3.80e11 + 3.19e11i)T^{2} \) |
| 97 | \( 1 + (-7.26e5 - 8.66e5i)T + (-1.44e11 + 8.20e11i)T^{2} \) |
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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
−13.75399144668249405776667034684, −12.58923106877738546949484806763, −11.12123344050107819298572836214, −10.42270991080873472317609091596, −9.486109010400986987285697232238, −7.70219833422314441436956725855, −6.71147859341077540900516673942, −4.91294686811064574982108338131, −3.86839578541471230631503321621, −1.79057212069668319170790224062,
0.61228314275240446449402340493, 2.06266639771574889557399967088, 4.24246860333556732182214925092, 5.79953138626933801838157992478, 6.84185848463111839769276074653, 8.527899198786718944022876444185, 9.146071941664058659955508487961, 10.97512796930441435194889640881, 12.01680743139255114860458804585, 12.72181906775693119636487207764
