| L(s) = 1 | + (5.87 + 2.13i)3-s + (−3.07 − 17.4i)5-s + (12.3 − 21.4i)7-s + (9.25 + 7.76i)9-s + (35.0 + 60.6i)11-s + (−19.4 + 7.08i)13-s + (19.2 − 108. i)15-s + (−3.03 + 2.54i)17-s + (−0.940 + 82.8i)19-s + (118. − 99.4i)21-s + (18.9 − 107. i)23-s + (−176. + 64.2i)25-s + (−46.6 − 80.7i)27-s + (80.6 + 67.7i)29-s + (−126. + 218. i)31-s + ⋯ |
| L(s) = 1 | + (1.13 + 0.411i)3-s + (−0.274 − 1.55i)5-s + (0.668 − 1.15i)7-s + (0.342 + 0.287i)9-s + (0.959 + 1.66i)11-s + (−0.415 + 0.151i)13-s + (0.330 − 1.87i)15-s + (−0.0433 + 0.0363i)17-s + (−0.0113 + 0.999i)19-s + (1.23 − 1.03i)21-s + (0.171 − 0.972i)23-s + (−1.41 + 0.514i)25-s + (−0.332 − 0.575i)27-s + (0.516 + 0.433i)29-s + (−0.730 + 1.26i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.852 + 0.522i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.852 + 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.99209 - 0.562177i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.99209 - 0.562177i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (0.940 - 82.8i)T \) |
| good | 3 | \( 1 + (-5.87 - 2.13i)T + (20.6 + 17.3i)T^{2} \) |
| 5 | \( 1 + (3.07 + 17.4i)T + (-117. + 42.7i)T^{2} \) |
| 7 | \( 1 + (-12.3 + 21.4i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (-35.0 - 60.6i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (19.4 - 7.08i)T + (1.68e3 - 1.41e3i)T^{2} \) |
| 17 | \( 1 + (3.03 - 2.54i)T + (853. - 4.83e3i)T^{2} \) |
| 23 | \( 1 + (-18.9 + 107. i)T + (-1.14e4 - 4.16e3i)T^{2} \) |
| 29 | \( 1 + (-80.6 - 67.7i)T + (4.23e3 + 2.40e4i)T^{2} \) |
| 31 | \( 1 + (126. - 218. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + 200.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-391. - 142. i)T + (5.27e4 + 4.43e4i)T^{2} \) |
| 43 | \( 1 + (8.35 + 47.3i)T + (-7.47e4 + 2.71e4i)T^{2} \) |
| 47 | \( 1 + (199. + 167. i)T + (1.80e4 + 1.02e5i)T^{2} \) |
| 53 | \( 1 + (72.6 - 411. i)T + (-1.39e5 - 5.09e4i)T^{2} \) |
| 59 | \( 1 + (-418. + 351. i)T + (3.56e4 - 2.02e5i)T^{2} \) |
| 61 | \( 1 + (27.7 - 157. i)T + (-2.13e5 - 7.76e4i)T^{2} \) |
| 67 | \( 1 + (-37.8 - 31.7i)T + (5.22e4 + 2.96e5i)T^{2} \) |
| 71 | \( 1 + (-83.4 - 473. i)T + (-3.36e5 + 1.22e5i)T^{2} \) |
| 73 | \( 1 + (734. + 267. i)T + (2.98e5 + 2.50e5i)T^{2} \) |
| 79 | \( 1 + (707. + 257. i)T + (3.77e5 + 3.16e5i)T^{2} \) |
| 83 | \( 1 + (167. - 290. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-367. + 133. i)T + (5.40e5 - 4.53e5i)T^{2} \) |
| 97 | \( 1 + (-449. + 377. i)T + (1.58e5 - 8.98e5i)T^{2} \) |
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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
−14.25465383951919224877977078599, −12.80781398147495323415982483048, −12.02542530811860156727859072032, −10.23994199404901095364357271158, −9.221721979492819574347971853008, −8.345929708685482300859798556824, −7.23491701130056302306327022769, −4.71767781813956397880847537827, −4.04218617055262860164946801122, −1.52010678709601315901788410946,
2.40023439046853997833255271201, 3.39633664509193602380381854658, 5.88180178875542203383363703396, 7.25738675693556553530943408474, 8.354409186510859223603717770133, 9.281649331043805185694880377030, 11.10536121752451007009224605668, 11.62884173410415216656729031755, 13.39250468300941933645277316824, 14.35148071073558469532037757875
