L(s) = 1 | + (0.907 + 0.243i)2-s + (1.12 + 1.31i)3-s + (−0.967 − 0.558i)4-s + (2.12 − 0.695i)5-s + (0.700 + 1.46i)6-s + (−2.64 + 0.0144i)7-s + (−2.07 − 2.07i)8-s + (−0.470 + 2.96i)9-s + (2.09 − 0.114i)10-s + (0.630 + 0.363i)11-s + (−0.352 − 1.90i)12-s + (−1.44 + 1.44i)13-s + (−2.40 − 0.630i)14-s + (3.30 + 2.01i)15-s + (−0.257 − 0.446i)16-s + (−1.90 − 7.09i)17-s + ⋯ |
L(s) = 1 | + (0.641 + 0.171i)2-s + (0.649 + 0.760i)3-s + (−0.483 − 0.279i)4-s + (0.950 − 0.311i)5-s + (0.285 + 0.599i)6-s + (−0.999 + 0.00544i)7-s + (−0.732 − 0.732i)8-s + (−0.156 + 0.987i)9-s + (0.663 − 0.0363i)10-s + (0.189 + 0.109i)11-s + (−0.101 − 0.549i)12-s + (−0.400 + 0.400i)13-s + (−0.642 − 0.168i)14-s + (0.853 + 0.520i)15-s + (−0.0644 − 0.111i)16-s + (−0.460 − 1.71i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 - 0.454i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.890 - 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.42112 + 0.341437i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.42112 + 0.341437i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.12 - 1.31i)T \) |
| 5 | \( 1 + (-2.12 + 0.695i)T \) |
| 7 | \( 1 + (2.64 - 0.0144i)T \) |
good | 2 | \( 1 + (-0.907 - 0.243i)T + (1.73 + i)T^{2} \) |
| 11 | \( 1 + (-0.630 - 0.363i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.44 - 1.44i)T - 13iT^{2} \) |
| 17 | \( 1 + (1.90 + 7.09i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-0.664 + 0.383i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.840 - 3.13i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 - 4.07T + 29T^{2} \) |
| 31 | \( 1 + (0.209 - 0.363i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.63 - 6.08i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + 4.44iT - 41T^{2} \) |
| 43 | \( 1 + (5.15 - 5.15i)T - 43iT^{2} \) |
| 47 | \( 1 + (-6.79 - 1.82i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-5.26 + 1.41i)T + (45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (-0.807 + 1.39i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.78 - 8.29i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.90 - 1.84i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 7.06iT - 71T^{2} \) |
| 73 | \( 1 + (4.08 + 15.2i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-5.80 + 3.35i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.83 - 1.83i)T + 83iT^{2} \) |
| 89 | \( 1 + (6.94 + 12.0i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.62 - 5.62i)T + 97iT^{2} \) |
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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
−13.64570571732165821770277944797, −13.45246608434879723439844771266, −12.03287200975879893526215087846, −10.19590425849805418294712634333, −9.517793184463093136823809587813, −8.921419409046556646751323945912, −6.86102560316005186398598892256, −5.48659013711093687890217826623, −4.47843809852646598585471751936, −2.91561138072949283237632359234,
2.50341200074928761404236028459, 3.76174904704881444174441147868, 5.76979049273608164238054647992, 6.72344818375139361284662092171, 8.335011247232703253377835206288, 9.267036483539657269055024943080, 10.40224376480948416405712851920, 12.22313725600981408911708607886, 12.86793691825964548245456133164, 13.53305011332370416329137340443