| L(s) = 1 | + (−8.89 − 1.37i)3-s + (27.4 + 15.8i)5-s + (37.6 + 65.2i)7-s + (77.2 + 24.4i)9-s + (−123. + 71.1i)11-s + (96.3 − 166. i)13-s + (−222. − 178. i)15-s + 325. i·17-s + 314.·19-s + (−245. − 632. i)21-s + (−443. − 256. i)23-s + (188. + 326. i)25-s + (−653. − 323. i)27-s + (−136. + 78.8i)29-s + (183. − 318. i)31-s + ⋯ |
| L(s) = 1 | + (−0.988 − 0.152i)3-s + (1.09 + 0.633i)5-s + (0.769 + 1.33i)7-s + (0.953 + 0.302i)9-s + (−1.01 + 0.588i)11-s + (0.569 − 0.987i)13-s + (−0.986 − 0.793i)15-s + 1.12i·17-s + 0.870·19-s + (−0.556 − 1.43i)21-s + (−0.839 − 0.484i)23-s + (0.301 + 0.522i)25-s + (−0.895 − 0.444i)27-s + (−0.162 + 0.0937i)29-s + (0.191 − 0.331i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.609 - 0.792i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.609 - 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.14457 + 0.563419i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.14457 + 0.563419i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (8.89 + 1.37i)T \) |
| good | 5 | \( 1 + (-27.4 - 15.8i)T + (312.5 + 541. i)T^{2} \) |
| 7 | \( 1 + (-37.6 - 65.2i)T + (-1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 + (123. - 71.1i)T + (7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 + (-96.3 + 166. i)T + (-1.42e4 - 2.47e4i)T^{2} \) |
| 17 | \( 1 - 325. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 314.T + 1.30e5T^{2} \) |
| 23 | \( 1 + (443. + 256. i)T + (1.39e5 + 2.42e5i)T^{2} \) |
| 29 | \( 1 + (136. - 78.8i)T + (3.53e5 - 6.12e5i)T^{2} \) |
| 31 | \( 1 + (-183. + 318. i)T + (-4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 - 1.73e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + (342. + 197. i)T + (1.41e6 + 2.44e6i)T^{2} \) |
| 43 | \( 1 + (360. + 624. i)T + (-1.70e6 + 2.96e6i)T^{2} \) |
| 47 | \( 1 + (-2.15e3 + 1.24e3i)T + (2.43e6 - 4.22e6i)T^{2} \) |
| 53 | \( 1 + 3.98e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + (2.13e3 + 1.23e3i)T + (6.05e6 + 1.04e7i)T^{2} \) |
| 61 | \( 1 + (1.24e3 + 2.14e3i)T + (-6.92e6 + 1.19e7i)T^{2} \) |
| 67 | \( 1 + (-3.29e3 + 5.71e3i)T + (-1.00e7 - 1.74e7i)T^{2} \) |
| 71 | \( 1 - 5.82e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 8.79e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + (-1.93e3 - 3.35e3i)T + (-1.94e7 + 3.37e7i)T^{2} \) |
| 83 | \( 1 + (-1.04e4 + 6.01e3i)T + (2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 - 7.63e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (-1.45e3 - 2.52e3i)T + (-4.42e7 + 7.66e7i)T^{2} \) |
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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
−15.78936617081254470159727961858, −14.86126284393026704365723522207, −13.29447000875158610658430142145, −12.23519487543969703324970536742, −10.87782157744949661459009351548, −9.941176561938831402422893595833, −8.011487137431332178138319136423, −6.10528594093418176205060390842, −5.31559502053140853306809283364, −2.12387168041844180199376090424,
1.14317894813591004663536020934, 4.57897114628499936074608384124, 5.80836490634411532696739157009, 7.49502464330555512343639667089, 9.472196982166778892568714448380, 10.62319944791907131824565854572, 11.66587745435343795028307577904, 13.39217644184165454911822771411, 13.90034695648760536664329819388, 16.03391523302799338555488569943
