| L(s) = 1 | + (−1.51 + 0.266i)2-s + (−1.68 + 2.48i)3-s + (−1.54 + 0.562i)4-s + (2.27 + 2.71i)5-s + (1.87 − 4.20i)6-s + (5.88 + 10.1i)7-s + (7.50 − 4.33i)8-s + (−3.34 − 8.35i)9-s + (−4.16 − 3.49i)10-s + 19.5i·11-s + (1.20 − 4.78i)12-s + (−14.5 − 12.2i)13-s + (−11.6 − 13.8i)14-s + (−10.5 + 1.09i)15-s + (−5.14 + 4.32i)16-s + (0.556 + 0.663i)17-s + ⋯ |
| L(s) = 1 | + (−0.755 + 0.133i)2-s + (−0.560 + 0.828i)3-s + (−0.386 + 0.140i)4-s + (0.455 + 0.543i)5-s + (0.313 − 0.700i)6-s + (0.840 + 1.45i)7-s + (0.937 − 0.541i)8-s + (−0.371 − 0.928i)9-s + (−0.416 − 0.349i)10-s + 1.77i·11-s + (0.100 − 0.398i)12-s + (−1.11 − 0.939i)13-s + (−0.828 − 0.987i)14-s + (−0.705 + 0.0730i)15-s + (−0.321 + 0.270i)16-s + (0.0327 + 0.0390i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0781i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.996 + 0.0781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0227661 - 0.581743i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0227661 - 0.581743i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.68 - 2.48i)T \) |
| 19 | \( 1 + (18.2 + 5.38i)T \) |
| good | 2 | \( 1 + (1.51 - 0.266i)T + (3.75 - 1.36i)T^{2} \) |
| 5 | \( 1 + (-2.27 - 2.71i)T + (-4.34 + 24.6i)T^{2} \) |
| 7 | \( 1 + (-5.88 - 10.1i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 - 19.5iT - 121T^{2} \) |
| 13 | \( 1 + (14.5 + 12.2i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (-0.556 - 0.663i)T + (-50.1 + 284. i)T^{2} \) |
| 23 | \( 1 + (1.40 + 3.86i)T + (-405. + 340. i)T^{2} \) |
| 29 | \( 1 + (-4.38 - 12.0i)T + (-644. + 540. i)T^{2} \) |
| 31 | \( 1 - 24.2T + 961T^{2} \) |
| 37 | \( 1 + 52.7T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-32.0 + 5.64i)T + (1.57e3 - 574. i)T^{2} \) |
| 43 | \( 1 + (32.4 + 11.8i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-15.4 - 42.4i)T + (-1.69e3 + 1.41e3i)T^{2} \) |
| 53 | \( 1 + (-43.4 - 7.65i)T + (2.63e3 + 960. i)T^{2} \) |
| 59 | \( 1 + (-17.9 + 49.2i)T + (-2.66e3 - 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-3.63 - 3.05i)T + (646. + 3.66e3i)T^{2} \) |
| 67 | \( 1 + (2.14 - 12.1i)T + (-4.21e3 - 1.53e3i)T^{2} \) |
| 71 | \( 1 + (32.3 - 5.70i)T + (4.73e3 - 1.72e3i)T^{2} \) |
| 73 | \( 1 + (-63.9 - 23.2i)T + (4.08e3 + 3.42e3i)T^{2} \) |
| 79 | \( 1 + (-30.3 + 25.5i)T + (1.08e3 - 6.14e3i)T^{2} \) |
| 83 | \( 1 + (90.3 - 52.1i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-24.2 - 66.5i)T + (-6.06e3 + 5.09e3i)T^{2} \) |
| 97 | \( 1 + (-25.3 - 143. i)T + (-8.84e3 + 3.21e3i)T^{2} \) |
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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
−12.59852766259835669979820139078, −12.10901438761297340743461636557, −10.61619658144323762023875362004, −10.00116026636016057552977094164, −9.167230556579361603952469260547, −8.159500227315585524283737066670, −6.82086823748392191244371732252, −5.29528963282388064261563723436, −4.55316037941829951013941237274, −2.36521369818220861723290834070,
0.50711287177466041430642940009, 1.65837608864189533783117452041, 4.41994515018706628451496121883, 5.48544889444261682445775448275, 6.95160750312912629795872803908, 8.019408740451394428281284872376, 8.775072980973591234754363206171, 10.16123204569050394988952343079, 10.92004974621794544321363349614, 11.77743503334975841290511603221
