| L(s) = 1 | + (−0.133 + 0.158i)2-s + (0.339 + 1.92i)4-s + (−1.06 + 1.96i)5-s + (−2.83 − 0.500i)7-s + (−0.711 − 0.410i)8-s + (−0.171 − 0.431i)10-s + (−1.38 + 0.502i)11-s + (−1.55 − 1.85i)13-s + (0.457 − 0.384i)14-s + (−3.51 + 1.28i)16-s + (−1.21 + 0.704i)17-s + (2.34 − 4.06i)19-s + (−4.15 − 1.37i)20-s + (0.104 − 0.286i)22-s + (−2.36 + 0.417i)23-s + ⋯ |
| L(s) = 1 | + (−0.0943 + 0.112i)2-s + (0.169 + 0.963i)4-s + (−0.474 + 0.880i)5-s + (−1.07 − 0.188i)7-s + (−0.251 − 0.145i)8-s + (−0.0542 − 0.136i)10-s + (−0.416 + 0.151i)11-s + (−0.431 − 0.514i)13-s + (0.122 − 0.102i)14-s + (−0.879 + 0.320i)16-s + (−0.295 + 0.170i)17-s + (0.538 − 0.931i)19-s + (−0.928 − 0.307i)20-s + (0.0222 − 0.0611i)22-s + (−0.493 + 0.0870i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.111i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 - 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0313271 + 0.562498i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0313271 + 0.562498i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (1.06 - 1.96i)T \) |
| good | 2 | \( 1 + (0.133 - 0.158i)T + (-0.347 - 1.96i)T^{2} \) |
| 7 | \( 1 + (2.83 + 0.500i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (1.38 - 0.502i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (1.55 + 1.85i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (1.21 - 0.704i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.34 + 4.06i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.36 - 0.417i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-6.73 - 5.65i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-1.00 - 5.72i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (7.57 - 4.37i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (8.32 - 6.98i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-2.63 - 7.23i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-6.68 - 1.17i)T + (44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 - 5.43iT - 53T^{2} \) |
| 59 | \( 1 + (6.83 + 2.48i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (1.03 - 5.89i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-4.95 - 5.90i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (4.51 + 7.82i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-10.8 - 6.28i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (1.14 + 0.963i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-7.46 + 8.89i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-5.96 + 10.3i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.92 + 5.28i)T + (-74.3 + 62.3i)T^{2} \) |
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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
−11.81052703915107803933612653300, −10.71755510252248488599272133597, −10.01442092070607361339597663249, −8.836312305103786532002000350309, −7.85494028939699800512313291587, −7.01037670246879153964740304619, −6.44477437131454475398438314669, −4.73360005155079582144188010431, −3.31753087608145562613341788350, −2.84167375897430021990163587036,
0.35425249709978298709618950835, 2.16993280458320582739305780116, 3.80175293418962151905308831030, 5.06819916191296148398577328511, 5.93617810265303888966409549161, 6.95280742784444506114526441903, 8.172276825351223036808467667677, 9.214074070199682746081904354968, 9.825822512627096697601017770588, 10.67449017311020456591356393594
