| L(s) = 1 | + (−2.30 − 3.00i)2-s + (1.06 + 2.80i)3-s + (−2.67 + 9.97i)4-s + (1.79 − 1.57i)5-s + (5.97 − 9.65i)6-s + (−7.21 − 6.33i)7-s + (22.1 − 9.16i)8-s + (−6.74 + 5.96i)9-s + (−8.84 − 1.75i)10-s + (0.768 + 11.7i)11-s + (−30.8 + 3.10i)12-s + (−15.8 − 4.25i)13-s + (−2.37 + 36.2i)14-s + (6.31 + 3.35i)15-s + (−42.7 − 24.6i)16-s + (−16.1 − 5.31i)17-s + ⋯ |
| L(s) = 1 | + (−1.15 − 1.50i)2-s + (0.354 + 0.935i)3-s + (−0.668 + 2.49i)4-s + (0.358 − 0.314i)5-s + (0.995 − 1.60i)6-s + (−1.03 − 0.904i)7-s + (2.76 − 1.14i)8-s + (−0.749 + 0.662i)9-s + (−0.884 − 0.175i)10-s + (0.0698 + 1.06i)11-s + (−2.56 + 0.258i)12-s + (−1.22 − 0.327i)13-s + (−0.169 + 2.59i)14-s + (0.420 + 0.223i)15-s + (−2.67 − 1.54i)16-s + (−0.949 − 0.312i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.262 - 0.965i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.262 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.100236 + 0.131078i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.100236 + 0.131078i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.06 - 2.80i)T \) |
| 17 | \( 1 + (16.1 + 5.31i)T \) |
| good | 2 | \( 1 + (2.30 + 3.00i)T + (-1.03 + 3.86i)T^{2} \) |
| 5 | \( 1 + (-1.79 + 1.57i)T + (3.26 - 24.7i)T^{2} \) |
| 7 | \( 1 + (7.21 + 6.33i)T + (6.39 + 48.5i)T^{2} \) |
| 11 | \( 1 + (-0.768 - 11.7i)T + (-119. + 15.7i)T^{2} \) |
| 13 | \( 1 + (15.8 + 4.25i)T + (146. + 84.5i)T^{2} \) |
| 19 | \( 1 + (10.8 - 26.1i)T + (-255. - 255. i)T^{2} \) |
| 23 | \( 1 + (-3.65 + 7.40i)T + (-322. - 419. i)T^{2} \) |
| 29 | \( 1 + (-6.77 + 19.9i)T + (-667. - 511. i)T^{2} \) |
| 31 | \( 1 + (0.305 - 4.65i)T + (-952. - 125. i)T^{2} \) |
| 37 | \( 1 + (15.7 - 23.5i)T + (-523. - 1.26e3i)T^{2} \) |
| 41 | \( 1 + (10.7 - 3.66i)T + (1.33e3 - 1.02e3i)T^{2} \) |
| 43 | \( 1 + (43.7 + 5.76i)T + (1.78e3 + 478. i)T^{2} \) |
| 47 | \( 1 + (7.30 + 27.2i)T + (-1.91e3 + 1.10e3i)T^{2} \) |
| 53 | \( 1 + (9.88 - 23.8i)T + (-1.98e3 - 1.98e3i)T^{2} \) |
| 59 | \( 1 + (-90.8 - 69.6i)T + (900. + 3.36e3i)T^{2} \) |
| 61 | \( 1 + (-35.5 + 40.5i)T + (-485. - 3.68e3i)T^{2} \) |
| 67 | \( 1 + (-41.4 + 23.9i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + (-11.7 + 17.5i)T + (-1.92e3 - 4.65e3i)T^{2} \) |
| 73 | \( 1 + (0.366 + 1.84i)T + (-4.92e3 + 2.03e3i)T^{2} \) |
| 79 | \( 1 + (5.03 + 76.8i)T + (-6.18e3 + 814. i)T^{2} \) |
| 83 | \( 1 + (-37.3 + 28.6i)T + (1.78e3 - 6.65e3i)T^{2} \) |
| 89 | \( 1 + (104. + 104. i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + (36.8 - 108. i)T + (-7.46e3 - 5.72e3i)T^{2} \) |
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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
−12.79173497550663980142245909050, −11.82506380475447368564930499419, −10.49704791887223839578070274617, −9.972375446801466000381401554281, −9.508679156073923817435654678374, −8.363738243868929988466987337255, −7.14418116665552311638464227359, −4.62706912341515823413677787811, −3.54214855766149871600686916592, −2.19468245554430901940978272037,
0.13546758659972673054225502660, 2.43459982171005860535107874263, 5.43642779440850130329428009641, 6.56839537898415626995459381199, 6.85291946064843214136733637760, 8.357508431392010643893237827492, 8.981377435499105263938717362512, 9.805169766331592518029490059152, 11.23195284253971800596247257685, 12.77154638797093509937288258511
