| L(s) = 1 | + (0.911 − 0.411i)2-s + (1.38 + 2.95i)3-s + (0.660 − 0.750i)4-s + (1.40 + 0.323i)5-s + (2.47 + 2.12i)6-s + (0.477 − 0.775i)7-s + (0.292 − 0.956i)8-s + (−4.88 + 5.88i)9-s + (1.41 − 0.284i)10-s + (0.868 − 4.66i)11-s + (3.13 + 0.911i)12-s + (1.22 − 0.103i)13-s + (0.115 − 0.902i)14-s + (0.991 + 4.60i)15-s + (−0.127 − 0.991i)16-s + (0.334 − 0.118i)17-s + ⋯ |
| L(s) = 1 | + (0.644 − 0.291i)2-s + (0.800 + 1.70i)3-s + (0.330 − 0.375i)4-s + (0.628 + 0.144i)5-s + (1.01 + 0.865i)6-s + (0.180 − 0.292i)7-s + (0.103 − 0.338i)8-s + (−1.62 + 1.96i)9-s + (0.447 − 0.0898i)10-s + (0.261 − 1.40i)11-s + (0.904 + 0.263i)12-s + (0.338 − 0.0288i)13-s + (0.0309 − 0.241i)14-s + (0.256 + 1.18i)15-s + (−0.0317 − 0.247i)16-s + (0.0810 − 0.0286i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 446 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.629 - 0.777i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 446 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.629 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.46111 + 1.17378i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.46111 + 1.17378i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.911 + 0.411i)T \) |
| 223 | \( 1 + (13.7 + 5.93i)T \) |
| good | 3 | \( 1 + (-1.38 - 2.95i)T + (-1.91 + 2.30i)T^{2} \) |
| 5 | \( 1 + (-1.40 - 0.323i)T + (4.49 + 2.18i)T^{2} \) |
| 7 | \( 1 + (-0.477 + 0.775i)T + (-3.15 - 6.25i)T^{2} \) |
| 11 | \( 1 + (-0.868 + 4.66i)T + (-10.2 - 3.95i)T^{2} \) |
| 13 | \( 1 + (-1.22 + 0.103i)T + (12.8 - 2.19i)T^{2} \) |
| 17 | \( 1 + (-0.334 + 0.118i)T + (13.2 - 10.6i)T^{2} \) |
| 19 | \( 1 + (2.51 - 0.142i)T + (18.8 - 2.14i)T^{2} \) |
| 23 | \( 1 + (4.34 - 4.66i)T + (-1.62 - 22.9i)T^{2} \) |
| 29 | \( 1 + (5.06 + 2.12i)T + (20.3 + 20.6i)T^{2} \) |
| 31 | \( 1 + (0.0852 - 0.543i)T + (-29.5 - 9.49i)T^{2} \) |
| 37 | \( 1 + (2.13 - 1.82i)T + (5.73 - 36.5i)T^{2} \) |
| 41 | \( 1 + (-0.390 + 9.19i)T + (-40.8 - 3.47i)T^{2} \) |
| 43 | \( 1 + (-0.0491 + 0.693i)T + (-42.5 - 6.06i)T^{2} \) |
| 47 | \( 1 + (-10.5 - 0.595i)T + (46.6 + 5.30i)T^{2} \) |
| 53 | \( 1 + (2.75 - 1.64i)T + (25.1 - 46.6i)T^{2} \) |
| 59 | \( 1 + (-8.94 - 8.57i)T + (2.50 + 58.9i)T^{2} \) |
| 61 | \( 1 + (3.29 + 4.72i)T + (-21.1 + 57.2i)T^{2} \) |
| 67 | \( 1 + (-13.2 - 2.65i)T + (61.8 + 25.8i)T^{2} \) |
| 71 | \( 1 + (-2.80 + 11.4i)T + (-62.9 - 32.8i)T^{2} \) |
| 73 | \( 1 + (4.99 + 11.4i)T + (-49.7 + 53.4i)T^{2} \) |
| 79 | \( 1 + (2.15 + 1.37i)T + (33.5 + 71.5i)T^{2} \) |
| 83 | \( 1 + (-0.820 - 11.5i)T + (-82.1 + 11.7i)T^{2} \) |
| 89 | \( 1 + (0.00522 - 0.00120i)T + (80.0 - 38.9i)T^{2} \) |
| 97 | \( 1 + (-7.59 - 8.15i)T + (-6.85 + 96.7i)T^{2} \) |
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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
−10.89392749224107653220261618309, −10.49066196950438446054785204531, −9.532171383907919495196871136112, −8.834961666879804745528322931132, −7.82110229860158292013685694131, −6.01809087368470147062118825279, −5.39702074470701610149322916695, −4.05253721110522691329221237112, −3.55375798797502584209933841306, −2.28275498602983542965837052243,
1.74995786692335558609186604937, 2.42581492744147872920988724931, 3.98621141113402848272879781816, 5.54706892505744723311970981279, 6.46051653715002080500033117218, 7.15310487237103149711836520023, 8.048051686188327312459388287044, 8.855072634587070707566523507615, 9.878473579270826105896827376257, 11.44228005972814422008798434377
