L(s) = 1 | + (1.80 + 1.04i)2-s + (1.38 + 1.04i)3-s + (1.17 + 2.03i)4-s + (1.41 + 3.32i)6-s + (−3.53 − 2.04i)7-s + 0.734i·8-s + (0.824 + 2.88i)9-s + (0.675 − 1.17i)11-s + (−0.498 + 4.04i)12-s + (−0.561 + 0.324i)13-s + (−4.26 − 7.38i)14-s + (1.58 − 2.74i)16-s − 1.35i·17-s + (−1.52 + 6.07i)18-s − 0.648·19-s + ⋯ |
L(s) = 1 | + (1.27 + 0.737i)2-s + (0.798 + 0.602i)3-s + (0.587 + 1.01i)4-s + (0.575 + 1.35i)6-s + (−1.33 − 0.772i)7-s + 0.259i·8-s + (0.274 + 0.961i)9-s + (0.203 − 0.353i)11-s + (−0.143 + 1.16i)12-s + (−0.155 + 0.0898i)13-s + (−1.13 − 1.97i)14-s + (0.396 − 0.686i)16-s − 0.327i·17-s + (−0.358 + 1.43i)18-s − 0.148·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.352 - 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.352 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.09112 + 1.44736i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.09112 + 1.44736i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.38 - 1.04i)T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (-1.80 - 1.04i)T + (1 + 1.73i)T^{2} \) |
| 7 | \( 1 + (3.53 + 2.04i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.675 + 1.17i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.561 - 0.324i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 1.35iT - 17T^{2} \) |
| 19 | \( 1 + 0.648T + 19T^{2} \) |
| 23 | \( 1 + (4.14 - 2.39i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.93 + 3.35i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.84 - 6.66i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 7.52iT - 37T^{2} \) |
| 41 | \( 1 + (-0.0898 - 0.155i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.710 + 0.410i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (9.44 + 5.45i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 4.17iT - 53T^{2} \) |
| 59 | \( 1 + (-2.08 - 3.61i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.91 + 3.30i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.05 + 4.07i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 6.11T + 71T^{2} \) |
| 73 | \( 1 - 12.3iT - 73T^{2} \) |
| 79 | \( 1 + (-5.17 + 8.95i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (10.6 + 6.12i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 3T + 89T^{2} \) |
| 97 | \( 1 + (-11.7 - 6.79i)T + (48.5 + 84.0i)T^{2} \) |
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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
−12.95624176630262810650774633253, −11.78489078013547450417519388939, −10.20757573526463969247941445780, −9.687181291793119590077432791550, −8.289469020947547280395295816625, −7.10320351464674741927789138793, −6.28687550158791771099418701198, −4.91730462906252309738136138183, −3.84658638109977286883962175326, −3.08241207758453772675010584640,
2.19133844309398681077792098830, 3.12378051812596816588524476749, 4.19526265774768943996549270892, 5.83783584372428733157870533656, 6.63610453864887129912941751275, 8.122048960976546467968110992571, 9.254632291797457169283805939309, 10.16932499626360270496101670655, 11.60798148861122611173817037066, 12.55317800538398844681250475820