| L(s) = 1 | + (−1.43 + 3.45i)2-s + (2.06 + 1.38i)3-s + (−7.07 − 7.07i)4-s + (1.12 − 0.223i)5-s + (−7.73 + 5.16i)6-s + (−5.36 − 1.06i)7-s + (20.7 − 8.59i)8-s + (−1.08 − 2.61i)9-s + (−0.835 + 4.20i)10-s + (−7.66 − 11.4i)11-s + (−4.84 − 24.3i)12-s + (−2.12 + 2.12i)13-s + (11.3 − 17.0i)14-s + (2.62 + 1.08i)15-s + 43.9i·16-s + ⋯ |
| L(s) = 1 | + (−0.715 + 1.72i)2-s + (0.688 + 0.460i)3-s + (−1.76 − 1.76i)4-s + (0.224 − 0.0446i)5-s + (−1.28 + 0.860i)6-s + (−0.766 − 0.152i)7-s + (2.59 − 1.07i)8-s + (−0.120 − 0.290i)9-s + (−0.0835 + 0.420i)10-s + (−0.696 − 1.04i)11-s + (−0.403 − 2.03i)12-s + (−0.163 + 0.163i)13-s + (0.812 − 1.21i)14-s + (0.175 + 0.0725i)15-s + 2.74i·16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0641i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.997 + 0.0641i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.631081 - 0.0202534i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.631081 - 0.0202534i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 + (1.43 - 3.45i)T + (-2.82 - 2.82i)T^{2} \) |
| 3 | \( 1 + (-2.06 - 1.38i)T + (3.44 + 8.31i)T^{2} \) |
| 5 | \( 1 + (-1.12 + 0.223i)T + (23.0 - 9.56i)T^{2} \) |
| 7 | \( 1 + (5.36 + 1.06i)T + (45.2 + 18.7i)T^{2} \) |
| 11 | \( 1 + (7.66 + 11.4i)T + (-46.3 + 111. i)T^{2} \) |
| 13 | \( 1 + (2.12 - 2.12i)T - 169iT^{2} \) |
| 19 | \( 1 + (-7.91 + 19.1i)T + (-255. - 255. i)T^{2} \) |
| 23 | \( 1 + (-14.3 + 9.58i)T + (202. - 488. i)T^{2} \) |
| 29 | \( 1 + (4.78 + 24.0i)T + (-776. + 321. i)T^{2} \) |
| 31 | \( 1 + (-2.89 + 4.33i)T + (-367. - 887. i)T^{2} \) |
| 37 | \( 1 + (-18.0 - 12.0i)T + (523. + 1.26e3i)T^{2} \) |
| 41 | \( 1 + (68.3 + 13.6i)T + (1.55e3 + 643. i)T^{2} \) |
| 43 | \( 1 + (-23.2 - 56.0i)T + (-1.30e3 + 1.30e3i)T^{2} \) |
| 47 | \( 1 + (50.5 - 50.5i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (0.756 - 1.82i)T + (-1.98e3 - 1.98e3i)T^{2} \) |
| 59 | \( 1 + (-21.6 + 8.97i)T + (2.46e3 - 2.46e3i)T^{2} \) |
| 61 | \( 1 + (-20.5 + 103. i)T + (-3.43e3 - 1.42e3i)T^{2} \) |
| 67 | \( 1 - 0.440iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (50.1 + 33.4i)T + (1.92e3 + 4.65e3i)T^{2} \) |
| 73 | \( 1 + (-34.6 + 6.89i)T + (4.92e3 - 2.03e3i)T^{2} \) |
| 79 | \( 1 + (2.51 + 3.75i)T + (-2.38e3 + 5.76e3i)T^{2} \) |
| 83 | \( 1 + (27.8 + 11.5i)T + (4.87e3 + 4.87e3i)T^{2} \) |
| 89 | \( 1 + (-17.0 - 17.0i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + (34.6 + 174. i)T + (-8.69e3 + 3.60e3i)T^{2} \) |
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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
−11.24316150991463836063479206101, −9.892135304127828812777026766678, −9.495468898073429441554651425172, −8.599810855739805280653451122680, −7.85013286472009018277468407980, −6.67180590676517144706398103598, −5.92046616070568855547143845124, −4.73630247953902449195492865233, −3.18218030051697978643211047777, −0.35937710445191528736759597965,
1.70560006020230141506821672046, 2.63488596769310938769041194994, 3.62080126084957411011989374918, 5.19740882391173000719021748127, 7.22131543584879949221579383483, 8.090491831709599836502661601156, 8.995885044028183646549423714935, 9.951415719642421298691825850680, 10.36407063697386597346012916655, 11.62465975221676589031209681204
