| L(s) = 1 | + (1.90 − 0.601i)2-s + (3.27 − 2.29i)4-s + (0.755 − 4.28i)5-s + (−6.44 − 7.67i)7-s + (4.86 − 6.34i)8-s + (−1.13 − 8.62i)10-s + (−16.5 + 2.92i)11-s + (7.70 − 2.80i)13-s + (−16.9 − 10.7i)14-s + (5.46 − 15.0i)16-s + (−13.2 + 22.9i)17-s + (10.2 − 5.92i)19-s + (−7.35 − 15.7i)20-s + (−29.8 + 15.5i)22-s + (14.3 − 17.1i)23-s + ⋯ |
| L(s) = 1 | + (0.953 − 0.300i)2-s + (0.818 − 0.573i)4-s + (0.151 − 0.856i)5-s + (−0.920 − 1.09i)7-s + (0.608 − 0.793i)8-s + (−0.113 − 0.862i)10-s + (−1.50 + 0.265i)11-s + (0.592 − 0.215i)13-s + (−1.20 − 0.769i)14-s + (0.341 − 0.939i)16-s + (−0.780 + 1.35i)17-s + (0.540 − 0.311i)19-s + (−0.367 − 0.788i)20-s + (−1.35 + 0.707i)22-s + (0.624 − 0.743i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.434 + 0.900i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.434 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.31540 - 2.09537i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.31540 - 2.09537i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.90 + 0.601i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-0.755 + 4.28i)T + (-23.4 - 8.55i)T^{2} \) |
| 7 | \( 1 + (6.44 + 7.67i)T + (-8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (16.5 - 2.92i)T + (113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (-7.70 + 2.80i)T + (129. - 108. i)T^{2} \) |
| 17 | \( 1 + (13.2 - 22.9i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-10.2 + 5.92i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-14.3 + 17.1i)T + (-91.8 - 520. i)T^{2} \) |
| 29 | \( 1 + (-29.1 - 10.6i)T + (644. + 540. i)T^{2} \) |
| 31 | \( 1 + (-12.9 + 15.4i)T + (-166. - 946. i)T^{2} \) |
| 37 | \( 1 + (-20.1 + 34.9i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-34.1 + 12.4i)T + (1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (-25.3 + 4.46i)T + (1.73e3 - 632. i)T^{2} \) |
| 47 | \( 1 + (-12.9 - 15.4i)T + (-383. + 2.17e3i)T^{2} \) |
| 53 | \( 1 + 46.7T + 2.80e3T^{2} \) |
| 59 | \( 1 + (36.4 + 6.42i)T + (3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (-48.7 + 40.9i)T + (646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (-25.9 - 71.3i)T + (-3.43e3 + 2.88e3i)T^{2} \) |
| 71 | \( 1 + (-17.4 - 10.0i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (50.2 + 87.0i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (29.7 - 81.8i)T + (-4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (25.2 - 69.3i)T + (-5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (-37.6 - 65.1i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-12.6 - 71.5i)T + (-8.84e3 + 3.21e3i)T^{2} \) |
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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
−10.84793145889659306664629953160, −10.57491994586065658990582670857, −9.462289945729466750935735398345, −8.133626147381823156229187433005, −7.00188381220009088434864937774, −6.01101167886424705958367092716, −4.89357627189957775929507377380, −3.97504393661200418375262281664, −2.66666757066473650324677956517, −0.854185230120330018215750975546,
2.70481303804256788362997487230, 3.01233987209381395037570661397, 4.84269827396336314882661742857, 5.83996510150745833085399517747, 6.61081315538556243386692808892, 7.59450528002638256126820612997, 8.781096901419132717238032603912, 9.983271371674296854394892146609, 11.01786975862891395840894991360, 11.75010708729201007017576923381
