L(s) = 1 | + (0.809 − 0.587i)2-s + (−1.02 + 0.334i)3-s + (0.309 − 0.951i)4-s + (−0.0854 − 0.0620i)5-s + (−0.636 + 0.875i)6-s + (−4.39 − 1.42i)7-s + (−0.309 − 0.951i)8-s + (−1.47 + 1.07i)9-s − 0.105·10-s + (1.40 − 3.00i)11-s + 1.08i·12-s + (−5.52 + 4.01i)13-s + (−4.39 + 1.42i)14-s + (0.108 + 0.0353i)15-s + (−0.809 − 0.587i)16-s + (3.34 − 4.59i)17-s + ⋯ |
L(s) = 1 | + (0.572 − 0.415i)2-s + (−0.594 + 0.193i)3-s + (0.154 − 0.475i)4-s + (−0.0381 − 0.0277i)5-s + (−0.259 + 0.357i)6-s + (−1.66 − 0.539i)7-s + (−0.109 − 0.336i)8-s + (−0.493 + 0.358i)9-s − 0.0333·10-s + (0.424 − 0.905i)11-s + 0.312i·12-s + (−1.53 + 1.11i)13-s + (−1.17 + 0.381i)14-s + (0.0280 + 0.00911i)15-s + (−0.202 − 0.146i)16-s + (0.810 − 1.11i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0244i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0244i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.00445175 + 0.364310i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00445175 + 0.364310i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.809 + 0.587i)T \) |
| 11 | \( 1 + (-1.40 + 3.00i)T \) |
| 19 | \( 1 + (2.45 - 3.60i)T \) |
good | 3 | \( 1 + (1.02 - 0.334i)T + (2.42 - 1.76i)T^{2} \) |
| 5 | \( 1 + (0.0854 + 0.0620i)T + (1.54 + 4.75i)T^{2} \) |
| 7 | \( 1 + (4.39 + 1.42i)T + (5.66 + 4.11i)T^{2} \) |
| 13 | \( 1 + (5.52 - 4.01i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-3.34 + 4.59i)T + (-5.25 - 16.1i)T^{2} \) |
| 23 | \( 1 + 2.16T + 23T^{2} \) |
| 29 | \( 1 + (0.338 - 1.04i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (3.76 + 5.18i)T + (-9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-4.02 - 1.30i)T + (29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-2.91 - 8.96i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 2.23iT - 43T^{2} \) |
| 47 | \( 1 + (1.52 + 4.69i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (5.55 + 7.64i)T + (-16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (2.70 + 0.880i)T + (47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (3.69 - 5.07i)T + (-18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + 8.17iT - 67T^{2} \) |
| 71 | \( 1 + (-3.22 + 4.44i)T + (-21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-7.79 - 2.53i)T + (59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-6.07 + 4.41i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-6.95 + 9.57i)T + (-25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 - 15.2iT - 89T^{2} \) |
| 97 | \( 1 + (4.85 + 6.68i)T + (-29.9 + 92.2i)T^{2} \) |
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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
−10.86328294214966036103316696050, −9.817617809297812676543355483470, −9.486849565022444668846481385053, −7.82872857814728623642824582237, −6.58488604985617806355343695864, −6.01408179659513093063075592595, −4.80486773704917342757183542090, −3.71002671232131879064613419691, −2.58673736581363998748085217114, −0.19150402716331169393683889052,
2.64781400646865872322401874305, 3.70208959226163677597230041980, 5.22628856948628695154364608300, 5.96221292873497281868691258755, 6.77821233453394918477596803736, 7.63321486423213802603881931735, 9.061400372119831382863071604087, 9.793584497710647953237588079439, 10.81149070950486657322074293229, 12.23220934864776213146687304640