| L(s) = 1 | + (0.844 + 1.13i)2-s + (−2.39 + 2.07i)3-s + (−0.573 + 1.91i)4-s + (1.26 − 0.181i)5-s + (−4.37 − 0.963i)6-s + (−0.503 + 1.10i)7-s + (−2.65 + 0.967i)8-s + (0.998 − 6.94i)9-s + (1.27 + 1.28i)10-s + (0.353 − 1.20i)11-s + (−2.59 − 5.77i)12-s + (−2.81 + 1.28i)13-s + (−1.67 + 0.359i)14-s + (−2.64 + 3.05i)15-s + (−3.34 − 2.19i)16-s + (5.48 + 3.52i)17-s + ⋯ |
| L(s) = 1 | + (0.597 + 0.802i)2-s + (−1.38 + 1.19i)3-s + (−0.286 + 0.957i)4-s + (0.565 − 0.0812i)5-s + (−1.78 − 0.393i)6-s + (−0.190 + 0.416i)7-s + (−0.939 + 0.341i)8-s + (0.332 − 2.31i)9-s + (0.402 + 0.404i)10-s + (0.106 − 0.362i)11-s + (−0.750 − 1.66i)12-s + (−0.779 + 0.356i)13-s + (−0.447 + 0.0961i)14-s + (−0.683 + 0.788i)15-s + (−0.835 − 0.549i)16-s + (1.33 + 0.855i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0476340 + 0.960996i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0476340 + 0.960996i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.844 - 1.13i)T \) |
| 23 | \( 1 + (0.666 - 4.74i)T \) |
| good | 3 | \( 1 + (2.39 - 2.07i)T + (0.426 - 2.96i)T^{2} \) |
| 5 | \( 1 + (-1.26 + 0.181i)T + (4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (0.503 - 1.10i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (-0.353 + 1.20i)T + (-9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (2.81 - 1.28i)T + (8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (-5.48 - 3.52i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (-1.39 - 2.17i)T + (-7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (0.390 - 0.607i)T + (-12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-4.40 + 5.08i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (-10.3 - 1.48i)T + (35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (-0.700 - 4.87i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (-3.42 + 2.97i)T + (6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + 4.61T + 47T^{2} \) |
| 53 | \( 1 + (13.0 + 5.95i)T + (34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (-9.20 + 4.20i)T + (38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (1.57 + 1.36i)T + (8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (-0.186 - 0.633i)T + (-56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (7.23 - 2.12i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (-9.55 + 6.13i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (5.85 + 12.8i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-8.85 - 1.27i)T + (79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (-1.74 - 2.01i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (1.43 + 10.0i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−12.98544072852218162364777520339, −12.02593096018077122558030678710, −11.40001191258434356895815339226, −9.902759794636068756131910206313, −9.462935915853575207182687433244, −7.81682391200921862048133656010, −6.14996962953361978181037562014, −5.79221044268389188545378090377, −4.75292862752207891611677522969, −3.54334037294227153946151617026,
0.891399910990127776485042645006, 2.51536854478527307106062666777, 4.74497156814896840063042054713, 5.66417063570257606718690157047, 6.60404703721536313750704264646, 7.63204672006219141235777781805, 9.707837696000790211395082484515, 10.41546162682956072109248206611, 11.46748411220059841802824016943, 12.22022573428652678431823226955
