L(s) = 1 | + (0.656 − 1.88i)2-s + (−3.13 − 2.48i)4-s + (3.27 − 3.77i)5-s − 9.55·7-s + (−6.74 + 4.29i)8-s + (−4.98 − 8.66i)10-s − 9.92i·11-s − 7.55i·13-s + (−6.27 + 18.0i)14-s + (3.68 + 15.5i)16-s − 17.1i·17-s + 26.1i·19-s + (−19.6 + 3.72i)20-s + (−18.7 − 6.51i)22-s − 1.67·23-s + ⋯ |
L(s) = 1 | + (0.328 − 0.944i)2-s + (−0.784 − 0.620i)4-s + (0.654 − 0.755i)5-s − 1.36·7-s + (−0.843 + 0.537i)8-s + (−0.498 − 0.866i)10-s − 0.902i·11-s − 0.581i·13-s + (−0.448 + 1.28i)14-s + (0.230 + 0.973i)16-s − 1.01i·17-s + 1.37i·19-s + (−0.982 + 0.186i)20-s + (−0.852 − 0.296i)22-s − 0.0728·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 + 0.186i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.982 + 0.186i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.118252 - 1.25758i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.118252 - 1.25758i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.656 + 1.88i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-3.27 + 3.77i)T \) |
good | 7 | \( 1 + 9.55T + 49T^{2} \) |
| 11 | \( 1 + 9.92iT - 121T^{2} \) |
| 13 | \( 1 + 7.55iT - 169T^{2} \) |
| 17 | \( 1 + 17.1iT - 289T^{2} \) |
| 19 | \( 1 - 26.1iT - 361T^{2} \) |
| 23 | \( 1 + 1.67T + 529T^{2} \) |
| 29 | \( 1 - 0.350T + 841T^{2} \) |
| 31 | \( 1 + 46.0iT - 961T^{2} \) |
| 37 | \( 1 - 22.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 77.2T + 1.68e3T^{2} \) |
| 43 | \( 1 - 41.7T + 1.84e3T^{2} \) |
| 47 | \( 1 - 14.0T + 2.20e3T^{2} \) |
| 53 | \( 1 + 22.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 94.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 38T + 3.72e3T^{2} \) |
| 67 | \( 1 - 29.8T + 4.48e3T^{2} \) |
| 71 | \( 1 + 7.19iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 34.3iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 46.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 24.1T + 6.88e3T^{2} \) |
| 89 | \( 1 + 100.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 131. iT - 9.40e3T^{2} \) |
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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
−12.20214542754859741842559882833, −10.99039405309674446770313887106, −9.810063867726946015925281268919, −9.433596479291980450979755135716, −8.176188680698868979469333691071, −6.18469673885978622491865905967, −5.48276417230821024269188304166, −3.90022570335255112118159639549, −2.64438373774194634118169222168, −0.67878948841024987211227666381,
2.78428631085804593766213294370, 4.18675573081047305299981795492, 5.74654539426743540502831202380, 6.67266785675550151066674257482, 7.26710672404418927578250338894, 8.986861305044358674383616802767, 9.643864760041608271648206529430, 10.73302720957361348054073684206, 12.36634269662077875089351448655, 13.02482094008775905069790712944