L(s) = 1 | + (0.0403 − 0.124i)2-s + (−1.72 + 0.115i)3-s + (1.60 + 1.16i)4-s + (0.951 − 0.309i)5-s + (−0.0553 + 0.219i)6-s + (−1.50 + 2.06i)7-s + (0.420 − 0.305i)8-s + (2.97 − 0.399i)9-s − 0.130i·10-s + (1.12 + 3.11i)11-s + (−2.90 − 1.82i)12-s + (4.12 + 1.34i)13-s + (0.196 + 0.270i)14-s + (−1.60 + 0.644i)15-s + (1.20 + 3.70i)16-s + (−1.85 − 5.72i)17-s + ⋯ |
L(s) = 1 | + (0.0285 − 0.0877i)2-s + (−0.997 + 0.0667i)3-s + (0.802 + 0.582i)4-s + (0.425 − 0.138i)5-s + (−0.0225 + 0.0894i)6-s + (−0.568 + 0.782i)7-s + (0.148 − 0.108i)8-s + (0.991 − 0.133i)9-s − 0.0412i·10-s + (0.339 + 0.940i)11-s + (−0.839 − 0.527i)12-s + (1.14 + 0.372i)13-s + (0.0524 + 0.0721i)14-s + (−0.415 + 0.166i)15-s + (0.301 + 0.926i)16-s + (−0.450 − 1.38i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.761 - 0.648i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.761 - 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.00568 + 0.370365i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.00568 + 0.370365i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.72 - 0.115i)T \) |
| 5 | \( 1 + (-0.951 + 0.309i)T \) |
| 11 | \( 1 + (-1.12 - 3.11i)T \) |
good | 2 | \( 1 + (-0.0403 + 0.124i)T + (-1.61 - 1.17i)T^{2} \) |
| 7 | \( 1 + (1.50 - 2.06i)T + (-2.16 - 6.65i)T^{2} \) |
| 13 | \( 1 + (-4.12 - 1.34i)T + (10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (1.85 + 5.72i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (0.284 + 0.391i)T + (-5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 - 4.22iT - 23T^{2} \) |
| 29 | \( 1 + (8.15 + 5.92i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-2.89 + 8.89i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (4.66 + 3.38i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (3.66 - 2.65i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 4.35iT - 43T^{2} \) |
| 47 | \( 1 + (0.504 + 0.694i)T + (-14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-8.89 - 2.89i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-4.80 + 6.60i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-4.13 + 1.34i)T + (49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 - 4.84T + 67T^{2} \) |
| 71 | \( 1 + (1.92 - 0.624i)T + (57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (0.412 - 0.568i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (9.67 + 3.14i)T + (63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (0.211 + 0.651i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 - 5.26iT - 89T^{2} \) |
| 97 | \( 1 + (-2.30 + 7.09i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−12.79110458705376646360842844317, −11.68130377152435694511654555071, −11.41122610532162875153857406810, −9.958641611350872596100300101193, −9.107104289770765232837882209575, −7.41190663458005036598652170227, −6.50038540573387716087985508226, −5.58394370784669155972409417757, −3.96981646553223542512315608591, −2.10661654235319421791046862727,
1.33739604845850678711214135498, 3.65340271887708245165605070629, 5.47623078821564545828232930682, 6.35036234489361944214588606423, 6.93008996735569478536736864708, 8.622334918494876384894228243318, 10.35327061244850215300019043762, 10.56041246100845174991633365747, 11.46677393166523895032744417714, 12.74721838987796998818134661702