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 + (−9.28 − 3.70i)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 + (−0.900 + 19.9i)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 + (−0.928 − 0.370i)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 + (−0.0450 + 0.998i)20-s + (0.852 − 0.296i)22-s + 0.0728·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0450 + 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0450 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.01264 - 1.05930i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.01264 - 1.05930i\) |
\(L(2)\) |
|
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
−12.07326101094153187450003034974, −11.22488509535935284925033864634, −10.20534653894084308889940183200, −9.227550572252362366142971042030, −8.411603657082906222746485080446, −7.29127173599130659856690748690, −5.14189127089276200382354193889, −4.61319451729424821952529536832, −2.52263218226953585578463943496, −1.18023235947412593160093871228,
1.71587304703013096225568445561, 4.04770244209767630608378260453, 5.55356571855635847609809054979, 6.28622600679925237112160574794, 7.65210848480699609956275395592, 8.383833551843684843797012810920, 9.567133495900019129078208960628, 10.63306437867562367626313373909, 11.37117247297276197975845410117, 12.99929583059927608695128501640