L(s) = 1 | + (−0.758 − 2.33i)2-s + (−0.809 + 0.587i)3-s + (−3.25 + 2.36i)4-s + (0.447 − 2.19i)5-s + (1.98 + 1.44i)6-s + 7-s + (4.00 + 2.91i)8-s + (0.309 − 0.951i)9-s + (−5.45 + 0.616i)10-s + (−1.26 − 3.88i)11-s + (1.24 − 3.82i)12-s + (−0.686 + 2.11i)13-s + (−0.758 − 2.33i)14-s + (0.925 + 2.03i)15-s + (1.27 − 3.91i)16-s + (−3.33 − 2.42i)17-s + ⋯ |
L(s) = 1 | + (−0.536 − 1.64i)2-s + (−0.467 + 0.339i)3-s + (−1.62 + 1.18i)4-s + (0.200 − 0.979i)5-s + (0.810 + 0.588i)6-s + 0.377·7-s + (1.41 + 1.02i)8-s + (0.103 − 0.317i)9-s + (−1.72 + 0.195i)10-s + (−0.380 − 1.17i)11-s + (0.358 − 1.10i)12-s + (−0.190 + 0.585i)13-s + (−0.202 − 0.623i)14-s + (0.239 + 0.525i)15-s + (0.317 − 0.977i)16-s + (−0.808 − 0.587i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.284 - 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.284 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.233368 + 0.312600i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.233368 + 0.312600i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.809 - 0.587i)T \) |
| 5 | \( 1 + (-0.447 + 2.19i)T \) |
| 7 | \( 1 - T \) |
good | 2 | \( 1 + (0.758 + 2.33i)T + (-1.61 + 1.17i)T^{2} \) |
| 11 | \( 1 + (1.26 + 3.88i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (0.686 - 2.11i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (3.33 + 2.42i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (5.18 + 3.76i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-0.955 - 2.94i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (0.990 - 0.719i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-5.36 - 3.89i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (2.12 - 6.54i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-2.14 + 6.58i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 5.93T + 43T^{2} \) |
| 47 | \( 1 + (7.12 - 5.17i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (6.26 - 4.55i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (2.21 - 6.82i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (2.93 + 9.02i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (-4.01 - 2.91i)T + (20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (-4.91 + 3.56i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (4.90 + 15.1i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-7.51 + 5.45i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-4.92 - 3.57i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (5.22 + 16.0i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-2.95 + 2.15i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.46068904747158327700396623318, −9.389625549773793077048965737504, −8.877236728064672543789311069303, −8.137025160754855277855307645147, −6.43452104446217761498582186499, −5.00217695249709954794561010212, −4.39516560922365855658454056454, −3.04398871478957756201509825466, −1.68635287333397500048875185250, −0.28563672581974136133891124791,
2.16350047204563566812595091231, 4.32482690793016977935733059387, 5.35977491914164605008899652740, 6.37557504133182146121622158543, 6.80613272324481698102443160916, 7.81956186507347127518340118110, 8.322786215861530101377510904846, 9.716458930169595593122478072088, 10.31771930373954187748801845070, 11.20173714713795555857680813624