L(s) = 1 | + (−0.5 + 0.866i)3-s + (1.5 + 2.59i)5-s + (−2.5 + 0.866i)7-s + (−0.499 − 0.866i)9-s + (−1.5 + 2.59i)11-s + 4·13-s − 3·15-s + (2 + 3.46i)19-s + (0.500 − 2.59i)21-s + (−2 + 3.46i)25-s + 0.999·27-s − 9·29-s + (−0.5 + 0.866i)31-s + (−1.5 − 2.59i)33-s + (−6 − 5.19i)35-s + ⋯ |
L(s) = 1 | + (−0.288 + 0.499i)3-s + (0.670 + 1.16i)5-s + (−0.944 + 0.327i)7-s + (−0.166 − 0.288i)9-s + (−0.452 + 0.783i)11-s + 1.10·13-s − 0.774·15-s + (0.458 + 0.794i)19-s + (0.109 − 0.566i)21-s + (−0.400 + 0.692i)25-s + 0.192·27-s − 1.67·29-s + (−0.0898 + 0.155i)31-s + (−0.261 − 0.452i)33-s + (−1.01 − 0.878i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.084396415\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.084396415\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (2.5 - 0.866i)T \) |
good | 5 | \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 9T + 29T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-4 - 6.92i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 10T + 43T^{2} \) |
| 47 | \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.5 - 2.59i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5 + 8.66i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5 + 8.66i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + (1 - 1.73i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 9T + 83T^{2} \) |
| 89 | \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + T + 97T^{2} \) |
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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
−9.847129592906555769621365340945, −9.617811925300647502603449069179, −8.460211171309675713827260941914, −7.37428904978834398234973024076, −6.48730825145589396331878131390, −6.00759532064066943862210745351, −5.10711785909175866657311552068, −3.71443981500927125131399719682, −3.08691029457206792552022307801, −1.88787079251985058206090425438,
0.45760126575825555831842628499, 1.55445830857693967280399678653, 2.97864268332861178571832407954, 4.06400188275481559516034203982, 5.33894426610144068278579162544, 5.80939920583384302638292252103, 6.64494406104385811027769021964, 7.62411236990218427561809685546, 8.566238297858644265930923755764, 9.191027185931119777766546501830