| L(s) = 1 | + (1.23 − 0.697i)2-s + (0.569 − 0.0560i)3-s + (1.02 − 1.71i)4-s + (−0.384 + 0.718i)5-s + (0.661 − 0.466i)6-s + (−0.174 + 0.877i)7-s + (0.0670 − 2.82i)8-s + (−2.62 + 0.521i)9-s + (0.0286 + 1.15i)10-s + (−3.03 + 2.48i)11-s + (0.488 − 1.03i)12-s + (4.70 − 2.51i)13-s + (0.397 + 1.20i)14-s + (−0.178 + 0.430i)15-s + (−1.88 − 3.52i)16-s + (1.22 + 2.96i)17-s + ⋯ |
| L(s) = 1 | + (0.869 − 0.493i)2-s + (0.328 − 0.0323i)3-s + (0.513 − 0.858i)4-s + (−0.171 + 0.321i)5-s + (0.269 − 0.190i)6-s + (−0.0659 + 0.331i)7-s + (0.0237 − 0.999i)8-s + (−0.873 + 0.173i)9-s + (0.00904 + 0.364i)10-s + (−0.914 + 0.750i)11-s + (0.141 − 0.298i)12-s + (1.30 − 0.697i)13-s + (0.106 + 0.321i)14-s + (−0.0460 + 0.111i)15-s + (−0.472 − 0.881i)16-s + (0.297 + 0.718i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.789 + 0.613i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.789 + 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.59488 - 0.546932i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.59488 - 0.546932i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.23 + 0.697i)T \) |
| good | 3 | \( 1 + (-0.569 + 0.0560i)T + (2.94 - 0.585i)T^{2} \) |
| 5 | \( 1 + (0.384 - 0.718i)T + (-2.77 - 4.15i)T^{2} \) |
| 7 | \( 1 + (0.174 - 0.877i)T + (-6.46 - 2.67i)T^{2} \) |
| 11 | \( 1 + (3.03 - 2.48i)T + (2.14 - 10.7i)T^{2} \) |
| 13 | \( 1 + (-4.70 + 2.51i)T + (7.22 - 10.8i)T^{2} \) |
| 17 | \( 1 + (-1.22 - 2.96i)T + (-12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (6.54 + 1.98i)T + (15.7 + 10.5i)T^{2} \) |
| 23 | \( 1 + (-2.16 - 1.44i)T + (8.80 + 21.2i)T^{2} \) |
| 29 | \( 1 + (-1.12 + 1.36i)T + (-5.65 - 28.4i)T^{2} \) |
| 31 | \( 1 + (0.934 + 0.934i)T + 31iT^{2} \) |
| 37 | \( 1 + (1.56 + 5.17i)T + (-30.7 + 20.5i)T^{2} \) |
| 41 | \( 1 + (-2.71 + 4.05i)T + (-15.6 - 37.8i)T^{2} \) |
| 43 | \( 1 + (-3.25 - 0.320i)T + (42.1 + 8.38i)T^{2} \) |
| 47 | \( 1 + (3.38 - 1.40i)T + (33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (-1.19 - 1.46i)T + (-10.3 + 51.9i)T^{2} \) |
| 59 | \( 1 + (3.03 + 1.62i)T + (32.7 + 49.0i)T^{2} \) |
| 61 | \( 1 + (0.772 + 7.84i)T + (-59.8 + 11.9i)T^{2} \) |
| 67 | \( 1 + (1.40 + 14.2i)T + (-65.7 + 13.0i)T^{2} \) |
| 71 | \( 1 + (-14.9 - 2.97i)T + (65.5 + 27.1i)T^{2} \) |
| 73 | \( 1 + (1.22 + 6.16i)T + (-67.4 + 27.9i)T^{2} \) |
| 79 | \( 1 + (-6.30 - 2.60i)T + (55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (0.363 - 1.19i)T + (-69.0 - 46.1i)T^{2} \) |
| 89 | \( 1 + (12.9 - 8.68i)T + (34.0 - 82.2i)T^{2} \) |
| 97 | \( 1 + (3.63 + 3.63i)T + 97iT^{2} \) |
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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
−13.10332967703367851712209815406, −12.51160760782395303779377513931, −11.00095117115502840999096376287, −10.67835174384505927948699073556, −9.085860407304370308787919297374, −7.84371044782386719570937205292, −6.31639196425178478939329472436, −5.26303744406987722786680221464, −3.64316830184442145585420110720, −2.38807060304961299325073386526,
2.89439389428426074727411602204, 4.20264497621472351277402245046, 5.63010080933898602521454869396, 6.68494163070814204465159647467, 8.235813346824665322238061380388, 8.702038097358591368067697538563, 10.68600969742033804208575538934, 11.53489084309948126999704904911, 12.71643049223969574550106353514, 13.61457410131945253262326314251
