| L(s) = 1 | + (−0.581 + 2.54i)2-s + (−4.34 − 2.09i)4-s + (−2.95 − 0.445i)5-s + (−0.339 + 0.588i)7-s + (4.59 − 5.76i)8-s + (2.85 − 7.26i)10-s + (1.98 − 0.955i)11-s + (−0.884 − 2.25i)13-s + (−1.30 − 1.20i)14-s + (5.99 + 7.52i)16-s + (2.49 − 0.376i)17-s + (−0.122 − 1.63i)19-s + (11.9 + 8.11i)20-s + (1.27 + 5.60i)22-s + (−0.0324 − 0.0221i)23-s + ⋯ |
| L(s) = 1 | + (−0.411 + 1.80i)2-s + (−2.17 − 1.04i)4-s + (−1.32 − 0.199i)5-s + (−0.128 + 0.222i)7-s + (1.62 − 2.03i)8-s + (0.901 − 2.29i)10-s + (0.597 − 0.287i)11-s + (−0.245 − 0.624i)13-s + (−0.347 − 0.322i)14-s + (1.49 + 1.88i)16-s + (0.606 − 0.0913i)17-s + (−0.0281 − 0.375i)19-s + (2.66 + 1.81i)20-s + (0.272 + 1.19i)22-s + (−0.00677 − 0.00461i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0740i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.449442 + 0.0166728i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.449442 + 0.0166728i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 43 | \( 1 + (6.51 + 0.781i)T \) |
| good | 2 | \( 1 + (0.581 - 2.54i)T + (-1.80 - 0.867i)T^{2} \) |
| 5 | \( 1 + (2.95 + 0.445i)T + (4.77 + 1.47i)T^{2} \) |
| 7 | \( 1 + (0.339 - 0.588i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.98 + 0.955i)T + (6.85 - 8.60i)T^{2} \) |
| 13 | \( 1 + (0.884 + 2.25i)T + (-9.52 + 8.84i)T^{2} \) |
| 17 | \( 1 + (-2.49 + 0.376i)T + (16.2 - 5.01i)T^{2} \) |
| 19 | \( 1 + (0.122 + 1.63i)T + (-18.7 + 2.83i)T^{2} \) |
| 23 | \( 1 + (0.0324 + 0.0221i)T + (8.40 + 21.4i)T^{2} \) |
| 29 | \( 1 + (5.57 + 5.17i)T + (2.16 + 28.9i)T^{2} \) |
| 31 | \( 1 + (-8.56 + 2.64i)T + (25.6 - 17.4i)T^{2} \) |
| 37 | \( 1 + (5.57 + 9.65i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.555 - 2.43i)T + (-36.9 - 17.7i)T^{2} \) |
| 47 | \( 1 + (7.92 + 3.81i)T + (29.3 + 36.7i)T^{2} \) |
| 53 | \( 1 + (-2.23 + 5.70i)T + (-38.8 - 36.0i)T^{2} \) |
| 59 | \( 1 + (2.97 + 3.73i)T + (-13.1 + 57.5i)T^{2} \) |
| 61 | \( 1 + (4.87 + 1.50i)T + (50.4 + 34.3i)T^{2} \) |
| 67 | \( 1 + (-0.194 - 2.58i)T + (-66.2 + 9.98i)T^{2} \) |
| 71 | \( 1 + (-7.71 + 5.25i)T + (25.9 - 66.0i)T^{2} \) |
| 73 | \( 1 + (1.43 + 3.66i)T + (-53.5 + 49.6i)T^{2} \) |
| 79 | \( 1 + (-3.10 + 5.37i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (3.97 - 3.68i)T + (6.20 - 82.7i)T^{2} \) |
| 89 | \( 1 + (7.30 - 6.77i)T + (6.65 - 88.7i)T^{2} \) |
| 97 | \( 1 + (0.852 - 0.410i)T + (60.4 - 75.8i)T^{2} \) |
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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
−11.37365509649942683536263708916, −10.01371340811329235677750565431, −9.118896960171705931243447665008, −8.196478885210137422715713095310, −7.71933875923257194763285633185, −6.77263024641163452227124423231, −5.76290578176307531402112225637, −4.72812735790989759097690079907, −3.65004617674706144675565647283, −0.37990153109547562408917015397,
1.44865887977271745971759853974, 3.12861593971888934681914238823, 3.87867196888982743613354329484, 4.79980064791319428817769735841, 6.87812610376496804504211787570, 7.981024413283155455256515108125, 8.765896608777494959707689003456, 9.794544761096936195808240673432, 10.49339843987754868043831998318, 11.51760229069087367875744884834
