| L(s) = 1 | + (−1.62 + 1.16i)2-s + (1.09 − 1.33i)3-s + (1.26 − 3.79i)4-s + (−0.522 − 1.72i)5-s + (−0.216 + 3.45i)6-s + (−1.31 + 6.59i)7-s + (2.38 + 7.63i)8-s + (−0.585 − 2.94i)9-s + (2.86 + 2.18i)10-s + (7.33 + 0.721i)11-s + (−3.69 − 5.86i)12-s + (−0.305 − 0.0927i)13-s + (−5.58 − 12.2i)14-s + (−2.87 − 1.19i)15-s + (−12.8 − 9.59i)16-s + (9.17 + 22.1i)17-s + ⋯ |
| L(s) = 1 | + (−0.811 + 0.584i)2-s + (0.366 − 0.446i)3-s + (0.316 − 0.948i)4-s + (−0.104 − 0.344i)5-s + (−0.0361 + 0.576i)6-s + (−0.187 + 0.942i)7-s + (0.298 + 0.954i)8-s + (−0.0650 − 0.326i)9-s + (0.286 + 0.218i)10-s + (0.666 + 0.0656i)11-s + (−0.307 − 0.488i)12-s + (−0.0235 − 0.00713i)13-s + (−0.399 − 0.874i)14-s + (−0.191 − 0.0794i)15-s + (−0.800 − 0.599i)16-s + (0.539 + 1.30i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.704 - 0.709i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.704 - 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.15286 + 0.479578i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.15286 + 0.479578i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.62 - 1.16i)T \) |
| 3 | \( 1 + (-1.09 + 1.33i)T \) |
| good | 5 | \( 1 + (0.522 + 1.72i)T + (-20.7 + 13.8i)T^{2} \) |
| 7 | \( 1 + (1.31 - 6.59i)T + (-45.2 - 18.7i)T^{2} \) |
| 11 | \( 1 + (-7.33 - 0.721i)T + (118. + 23.6i)T^{2} \) |
| 13 | \( 1 + (0.305 + 0.0927i)T + (140. + 93.8i)T^{2} \) |
| 17 | \( 1 + (-9.17 - 22.1i)T + (-204. + 204. i)T^{2} \) |
| 19 | \( 1 + (19.1 - 10.2i)T + (200. - 300. i)T^{2} \) |
| 23 | \( 1 + (-8.50 - 5.68i)T + (202. + 488. i)T^{2} \) |
| 29 | \( 1 + (-40.3 + 3.97i)T + (824. - 164. i)T^{2} \) |
| 31 | \( 1 + (-33.0 + 33.0i)T - 961iT^{2} \) |
| 37 | \( 1 + (59.6 + 31.8i)T + (760. + 1.13e3i)T^{2} \) |
| 41 | \( 1 + (-62.1 - 41.5i)T + (643. + 1.55e3i)T^{2} \) |
| 43 | \( 1 + (-49.9 - 60.9i)T + (-360. + 1.81e3i)T^{2} \) |
| 47 | \( 1 + (-21.4 - 51.8i)T + (-1.56e3 + 1.56e3i)T^{2} \) |
| 53 | \( 1 + (46.5 + 4.58i)T + (2.75e3 + 548. i)T^{2} \) |
| 59 | \( 1 + (-3.92 - 12.9i)T + (-2.89e3 + 1.93e3i)T^{2} \) |
| 61 | \( 1 + (-20.3 + 24.7i)T + (-725. - 3.64e3i)T^{2} \) |
| 67 | \( 1 + (2.12 + 1.74i)T + (875. + 4.40e3i)T^{2} \) |
| 71 | \( 1 + (-56.9 - 11.3i)T + (4.65e3 + 1.92e3i)T^{2} \) |
| 73 | \( 1 + (92.6 - 18.4i)T + (4.92e3 - 2.03e3i)T^{2} \) |
| 79 | \( 1 + (3.55 - 8.58i)T + (-4.41e3 - 4.41e3i)T^{2} \) |
| 83 | \( 1 + (-6.03 - 11.2i)T + (-3.82e3 + 5.72e3i)T^{2} \) |
| 89 | \( 1 + (58.6 + 87.7i)T + (-3.03e3 + 7.31e3i)T^{2} \) |
| 97 | \( 1 + (-110. - 110. i)T + 9.40e3iT^{2} \) |
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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
−11.10694516291364651012328979496, −10.06695836989696154888269291085, −9.113295972460647442212083715972, −8.472899828949941337734134410010, −7.76330563798038274262184836885, −6.40141844619921048828482177818, −5.95266154563724379823549143241, −4.40851549009490714140586712281, −2.56583860058452556934462188130, −1.21366846550437807702354534495,
0.838248522607478089386768381802, 2.67095366434449284582723980319, 3.64723653633119933182365166131, 4.73410483114856799814332949702, 6.76878092958065915193907885205, 7.24932423418430739644798052314, 8.566040285071599992004210217612, 9.158689249768397297963270487268, 10.35878969765705175951274499577, 10.58379826318613403421379698500
