| L(s) = 1 | + (−0.0990 + 1.41i)2-s + (−2.04 − 1.09i)3-s + (−1.98 − 0.279i)4-s + (1.40 − 1.71i)5-s + (1.74 − 2.77i)6-s + (−3.76 − 2.51i)7-s + (0.590 − 2.76i)8-s + (1.32 + 1.97i)9-s + (2.28 + 2.15i)10-s + (2.34 + 0.711i)11-s + (3.74 + 2.73i)12-s + (0.439 − 0.360i)13-s + (3.92 − 5.06i)14-s + (−4.75 + 1.96i)15-s + (3.84 + 1.10i)16-s + (−7.29 − 3.01i)17-s + ⋯ |
| L(s) = 1 | + (−0.0700 + 0.997i)2-s + (−1.18 − 0.631i)3-s + (−0.990 − 0.139i)4-s + (0.629 − 0.767i)5-s + (0.712 − 1.13i)6-s + (−1.42 − 0.951i)7-s + (0.208 − 0.977i)8-s + (0.440 + 0.659i)9-s + (0.721 + 0.681i)10-s + (0.707 + 0.214i)11-s + (1.08 + 0.789i)12-s + (0.121 − 0.0999i)13-s + (1.04 − 1.35i)14-s + (−1.22 + 0.508i)15-s + (0.960 + 0.276i)16-s + (−1.76 − 0.732i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.243 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.243 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.385855 - 0.301090i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.385855 - 0.301090i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.0990 - 1.41i)T \) |
| good | 3 | \( 1 + (2.04 + 1.09i)T + (1.66 + 2.49i)T^{2} \) |
| 5 | \( 1 + (-1.40 + 1.71i)T + (-0.975 - 4.90i)T^{2} \) |
| 7 | \( 1 + (3.76 + 2.51i)T + (2.67 + 6.46i)T^{2} \) |
| 11 | \( 1 + (-2.34 - 0.711i)T + (9.14 + 6.11i)T^{2} \) |
| 13 | \( 1 + (-0.439 + 0.360i)T + (2.53 - 12.7i)T^{2} \) |
| 17 | \( 1 + (7.29 + 3.01i)T + (12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (-0.155 + 1.58i)T + (-18.6 - 3.70i)T^{2} \) |
| 23 | \( 1 + (-1.54 - 0.307i)T + (21.2 + 8.80i)T^{2} \) |
| 29 | \( 1 + (1.22 + 4.04i)T + (-24.1 + 16.1i)T^{2} \) |
| 31 | \( 1 + (0.936 - 0.936i)T - 31iT^{2} \) |
| 37 | \( 1 + (-7.54 + 0.742i)T + (36.2 - 7.21i)T^{2} \) |
| 41 | \( 1 + (0.504 - 2.53i)T + (-37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (-8.44 + 4.51i)T + (23.8 - 35.7i)T^{2} \) |
| 47 | \( 1 + (-4.89 + 11.8i)T + (-33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (0.556 - 1.83i)T + (-44.0 - 29.4i)T^{2} \) |
| 59 | \( 1 + (0.239 + 0.196i)T + (11.5 + 57.8i)T^{2} \) |
| 61 | \( 1 + (-4.32 + 8.09i)T + (-33.8 - 50.7i)T^{2} \) |
| 67 | \( 1 + (3.98 - 7.45i)T + (-37.2 - 55.7i)T^{2} \) |
| 71 | \( 1 + (-3.92 + 5.87i)T + (-27.1 - 65.5i)T^{2} \) |
| 73 | \( 1 + (-2.97 + 1.98i)T + (27.9 - 67.4i)T^{2} \) |
| 79 | \( 1 + (-4.13 - 9.98i)T + (-55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (12.8 + 1.26i)T + (81.4 + 16.1i)T^{2} \) |
| 89 | \( 1 + (8.22 - 1.63i)T + (82.2 - 34.0i)T^{2} \) |
| 97 | \( 1 + (6.77 - 6.77i)T - 97iT^{2} \) |
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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
−13.21347583152803976836075824877, −12.53236868493002222886986293738, −11.08834815411471117899949463838, −9.713324137032863578581283144181, −9.009440427237685876150435006247, −7.10897759578172760229186626797, −6.60695731050975367655525749430, −5.61783636833385333756632455720, −4.27385850052198812017576252329, −0.60796130462897932007799359271,
2.63537221696896834216500824767, 4.18201705844546065011157259445, 5.82240110366562181185989419961, 6.44061303081744903659447363683, 8.936162579583504461405596593808, 9.656832481397589244957665683240, 10.65554710002516906226783257732, 11.27314478408238533634388090503, 12.38171633537278039416909301276, 13.16469637221941691274760147595
