L(s) = 1 | + (0.866 − 1.5i)2-s + (−1 + 1.73i)3-s + (−0.5 − 0.866i)4-s + 1.73·5-s + (1.73 + 3i)6-s + 1.73·8-s + (−0.499 − 0.866i)9-s + (1.49 − 2.59i)10-s + 2·12-s + (−1.73 + 2.99i)15-s + (2.49 − 4.33i)16-s + (−1.5 − 2.59i)17-s − 1.73·18-s + (−1.73 − 3i)19-s + (−0.866 − 1.50i)20-s + ⋯ |
L(s) = 1 | + (0.612 − 1.06i)2-s + (−0.577 + 0.999i)3-s + (−0.250 − 0.433i)4-s + 0.774·5-s + (0.707 + 1.22i)6-s + 0.612·8-s + (−0.166 − 0.288i)9-s + (0.474 − 0.821i)10-s + 0.577·12-s + (−0.447 + 0.774i)15-s + (0.624 − 1.08i)16-s + (−0.363 − 0.630i)17-s − 0.408·18-s + (−0.397 − 0.688i)19-s + (−0.193 − 0.335i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.48562 - 0.249374i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.48562 - 0.249374i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 \) |
good | 2 | \( 1 + (-0.866 + 1.5i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (1 - 1.73i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 - 1.73T + 5T^{2} \) |
| 7 | \( 1 + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.73 + 3i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 3.46T + 31T^{2} \) |
| 37 | \( 1 + (-4.33 + 7.5i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.59 + 4.5i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-4 - 6.92i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 3.46T + 47T^{2} \) |
| 53 | \( 1 + 3T + 53T^{2} \) |
| 59 | \( 1 + (3.46 + 6i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.73 - 3i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.73 - 3i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 1.73T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 + (3.46 - 6i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.46 - 6i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−12.69852044390338259401356996801, −11.44469772120690483190998004542, −10.98050381169870328059696612846, −9.984192833419730653920066752652, −9.293738423359049310958431943775, −7.45145844507463007729883656062, −5.78011015459741765677317590653, −4.82934182953175803873413901625, −3.75789500024280062620500480945, −2.16477055288425466478804862196,
1.80752134955969724882689588861, 4.34192001417676835383240037133, 5.91962893181555972810241465281, 6.15359377268972769751707187780, 7.25815811685853860469678321852, 8.276433748151106426137030070353, 9.866519652568439955142800669401, 10.94969145802162550966834362624, 12.26874644434234748299709762454, 13.03113597301930992250082490013