| L(s) = 1 | + (0.981 − 0.189i)2-s + (−1.27 + 2.78i)3-s + (0.928 − 0.371i)4-s + (−0.0238 − 0.00701i)5-s + (−0.722 + 2.97i)6-s + (−0.733 + 2.11i)7-s + (0.841 − 0.540i)8-s + (−4.19 − 4.83i)9-s + (−0.0247 − 0.00236i)10-s + (0.377 + 1.55i)11-s + (−0.145 + 3.06i)12-s + (4.40 − 2.27i)13-s + (−0.319 + 2.21i)14-s + (0.0499 − 0.0576i)15-s + (0.723 − 0.690i)16-s + (5.31 + 2.12i)17-s + ⋯ |
| L(s) = 1 | + (0.694 − 0.133i)2-s + (−0.735 + 1.61i)3-s + (0.464 − 0.185i)4-s + (−0.0106 − 0.00313i)5-s + (−0.295 + 1.21i)6-s + (−0.277 + 0.800i)7-s + (0.297 − 0.191i)8-s + (−1.39 − 1.61i)9-s + (−0.00783 − 0.000748i)10-s + (0.113 + 0.469i)11-s + (−0.0421 + 0.883i)12-s + (1.22 − 0.629i)13-s + (−0.0852 + 0.593i)14-s + (0.0129 − 0.0148i)15-s + (0.180 − 0.172i)16-s + (1.28 + 0.515i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.217 - 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.217 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.986781 + 0.790903i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.986781 + 0.790903i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.981 + 0.189i)T \) |
| 67 | \( 1 + (7.49 + 3.29i)T \) |
| good | 3 | \( 1 + (1.27 - 2.78i)T + (-1.96 - 2.26i)T^{2} \) |
| 5 | \( 1 + (0.0238 + 0.00701i)T + (4.20 + 2.70i)T^{2} \) |
| 7 | \( 1 + (0.733 - 2.11i)T + (-5.50 - 4.32i)T^{2} \) |
| 11 | \( 1 + (-0.377 - 1.55i)T + (-9.77 + 5.04i)T^{2} \) |
| 13 | \( 1 + (-4.40 + 2.27i)T + (7.54 - 10.5i)T^{2} \) |
| 17 | \( 1 + (-5.31 - 2.12i)T + (12.3 + 11.7i)T^{2} \) |
| 19 | \( 1 + (1.95 + 5.63i)T + (-14.9 + 11.7i)T^{2} \) |
| 23 | \( 1 + (0.243 + 0.341i)T + (-7.52 + 21.7i)T^{2} \) |
| 29 | \( 1 + (2.62 + 4.55i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-6.33 - 3.26i)T + (17.9 + 25.2i)T^{2} \) |
| 37 | \( 1 + (-0.665 + 1.15i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (9.22 - 7.25i)T + (9.66 - 39.8i)T^{2} \) |
| 43 | \( 1 + (0.663 + 4.61i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + (5.10 - 0.487i)T + (46.1 - 8.89i)T^{2} \) |
| 53 | \( 1 + (1.08 - 7.52i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (-10.8 + 6.95i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-0.272 + 1.12i)T + (-54.2 - 27.9i)T^{2} \) |
| 71 | \( 1 + (14.0 - 5.62i)T + (51.3 - 48.9i)T^{2} \) |
| 73 | \( 1 + (-1.04 + 4.29i)T + (-64.8 - 33.4i)T^{2} \) |
| 79 | \( 1 + (0.137 - 2.88i)T + (-78.6 - 7.50i)T^{2} \) |
| 83 | \( 1 + (-0.262 + 0.250i)T + (3.94 - 82.9i)T^{2} \) |
| 89 | \( 1 + (-3.24 - 7.10i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (-9.46 + 16.4i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−13.43353839134972561539711307071, −12.20302983359022426724671143349, −11.45555933481833605566783684644, −10.44901036039939233949060415226, −9.701005734465376921856910357347, −8.443495011587687167826951198903, −6.28645678469765769121134615029, −5.52137765348529071208288682157, −4.38147046503012587900468461728, −3.20202798195284750499162825113,
1.44626389671974349127342455815, 3.62943186355133798851063448809, 5.58324684313531677167621200105, 6.38892800246173129021276741671, 7.33147390484040595213458170821, 8.288342815460611001783172521269, 10.31690559795323291785313725639, 11.51591861089335651786830537916, 12.03551993276212465991674398604, 13.26136378899067053164433690569
