L(s) = 1 | − i·2-s + (0.618 − 1.61i)3-s − 4-s + 1.23·5-s + (−1.61 − 0.618i)6-s + (2.61 + 0.381i)7-s + i·8-s + (−2.23 − 2.00i)9-s − 1.23i·10-s + i·11-s + (−0.618 + 1.61i)12-s − 6i·13-s + (0.381 − 2.61i)14-s + (0.763 − 2.00i)15-s + 16-s + 2.76·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (0.356 − 0.934i)3-s − 0.5·4-s + 0.552·5-s + (−0.660 − 0.252i)6-s + (0.989 + 0.144i)7-s + 0.353i·8-s + (−0.745 − 0.666i)9-s − 0.390i·10-s + 0.301i·11-s + (−0.178 + 0.467i)12-s − 1.66i·13-s + (0.102 − 0.699i)14-s + (0.197 − 0.516i)15-s + 0.250·16-s + 0.670·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.487 + 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.487 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.871863 - 1.48623i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.871863 - 1.48623i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + (-0.618 + 1.61i)T \) |
| 7 | \( 1 + (-2.61 - 0.381i)T \) |
| 11 | \( 1 - iT \) |
good | 5 | \( 1 - 1.23T + 5T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 - 2.76T + 17T^{2} \) |
| 19 | \( 1 + 2iT - 19T^{2} \) |
| 23 | \( 1 - 3.23iT - 23T^{2} \) |
| 29 | \( 1 - 8.47iT - 29T^{2} \) |
| 31 | \( 1 + 5.23iT - 31T^{2} \) |
| 37 | \( 1 - 4.47T + 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 + 6.94T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 - 5.70iT - 53T^{2} \) |
| 59 | \( 1 + 1.23T + 59T^{2} \) |
| 61 | \( 1 - 12.4iT - 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 + 16.1iT - 71T^{2} \) |
| 73 | \( 1 - 8.18iT - 73T^{2} \) |
| 79 | \( 1 - 2.76T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 - 4.94T + 89T^{2} \) |
| 97 | \( 1 + 6.47iT - 97T^{2} \) |
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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
−10.82427005422884450355433478564, −9.938996810374542909397110885630, −8.917263915757782937350626342501, −8.043201032867115936847951399771, −7.36203227983659590319131766139, −5.85051485121645107465174064908, −5.12833905337014648757307953939, −3.43125204184876553210421189297, −2.31279278911285762268133530575, −1.17292470199172224556447731944,
2.02024850973126980547445189196, 3.79730004063956671084240800959, 4.67859075981237580197284009778, 5.58059461009239174623110925924, 6.64786886128596789253957281411, 7.946442717532163595655263088628, 8.571277210422102481087504670426, 9.531146501281118413878703356241, 10.17403937152982549605093155466, 11.26193858753794294011789602936