L(s) = 1 | − 1.73i·7-s + 3.46i·11-s + (−4.5 + 2.59i)13-s + (3 − 5.19i)17-s + (−4 − 1.73i)19-s + (3 − 1.73i)23-s + (2.5 + 4.33i)25-s + (3 − 1.73i)29-s + 31-s − 8.66i·37-s + (6 + 3.46i)41-s + (−4.5 − 2.59i)43-s + (9 − 5.19i)47-s + 4·49-s + (9 − 5.19i)53-s + ⋯ |
L(s) = 1 | − 0.654i·7-s + 1.04i·11-s + (−1.24 + 0.720i)13-s + (0.727 − 1.26i)17-s + (−0.917 − 0.397i)19-s + (0.625 − 0.361i)23-s + (0.5 + 0.866i)25-s + (0.557 − 0.321i)29-s + 0.179·31-s − 1.42i·37-s + (0.937 + 0.541i)41-s + (−0.686 − 0.396i)43-s + (1.31 − 0.757i)47-s + 0.571·49-s + (1.23 − 0.713i)53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.671 + 0.740i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.671 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.550413757\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.550413757\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (4 + 1.73i)T \) |
good | 5 | \( 1 + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + 1.73iT - 7T^{2} \) |
| 11 | \( 1 - 3.46iT - 11T^{2} \) |
| 13 | \( 1 + (4.5 - 2.59i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-3 + 1.73i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3 + 1.73i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - T + 31T^{2} \) |
| 37 | \( 1 + 8.66iT - 37T^{2} \) |
| 41 | \( 1 + (-6 - 3.46i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.5 + 2.59i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-9 + 5.19i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-9 + 5.19i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (6 - 10.3i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.5 - 4.33i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.5 + 11.2i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.5 + 4.33i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 17.3iT - 83T^{2} \) |
| 89 | \( 1 + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-12 - 6.92i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.014214153243778854867179585060, −7.66350171950873883741614778833, −7.21966101702330435056828786085, −6.75174274607511829942234768190, −5.48002251931227763329425085367, −4.69493644329183855329332637383, −4.16682532921498144871198831288, −2.86831235986725589289400852387, −2.05495924288036352418961036127, −0.60542505835964058463003449133,
0.998996254448405603439831032123, 2.41461399522982433159967505647, 3.11494196354921670215982180470, 4.17151558810655660165013039338, 5.16546840831288235276245419748, 5.84211032669462278804462405690, 6.49430988263262441817710785932, 7.53848099665972617328565541188, 8.327453372297320095684435356319, 8.687060378136006594473080065015