| L(s) = 1 | + (0.997 − 0.0713i)2-s + (−0.664 + 3.05i)3-s + (0.989 − 0.142i)4-s + (2.09 + 0.779i)5-s + (−0.444 + 3.09i)6-s + (1.24 − 2.28i)7-s + (0.977 − 0.212i)8-s + (−6.15 − 2.81i)9-s + (2.14 + 0.628i)10-s + (−3.47 + 3.01i)11-s + (−0.222 + 3.11i)12-s + (2.28 − 1.24i)13-s + (1.08 − 2.36i)14-s + (−3.77 + 5.88i)15-s + (0.959 − 0.281i)16-s + (−2.24 − 2.99i)17-s + ⋯ |
| L(s) = 1 | + (0.705 − 0.0504i)2-s + (−0.383 + 1.76i)3-s + (0.494 − 0.0711i)4-s + (0.937 + 0.348i)5-s + (−0.181 + 1.26i)6-s + (0.471 − 0.863i)7-s + (0.345 − 0.0751i)8-s + (−2.05 − 0.937i)9-s + (0.678 + 0.198i)10-s + (−1.04 + 0.908i)11-s + (−0.0643 + 0.899i)12-s + (0.633 − 0.346i)13-s + (0.288 − 0.632i)14-s + (−0.974 + 1.51i)15-s + (0.239 − 0.0704i)16-s + (−0.544 − 0.727i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.204 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.204 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.39847 + 1.13615i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.39847 + 1.13615i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.997 + 0.0713i)T \) |
| 5 | \( 1 + (-2.09 - 0.779i)T \) |
| 23 | \( 1 + (4.73 + 0.785i)T \) |
| good | 3 | \( 1 + (0.664 - 3.05i)T + (-2.72 - 1.24i)T^{2} \) |
| 7 | \( 1 + (-1.24 + 2.28i)T + (-3.78 - 5.88i)T^{2} \) |
| 11 | \( 1 + (3.47 - 3.01i)T + (1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-2.28 + 1.24i)T + (7.02 - 10.9i)T^{2} \) |
| 17 | \( 1 + (2.24 + 2.99i)T + (-4.78 + 16.3i)T^{2} \) |
| 19 | \( 1 + (0.215 + 1.49i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (-6.04 - 0.869i)T + (27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (-4.50 - 2.89i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (3.04 + 8.17i)T + (-27.9 + 24.2i)T^{2} \) |
| 41 | \( 1 + (4.39 + 9.61i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-0.661 - 0.143i)T + (39.1 + 17.8i)T^{2} \) |
| 47 | \( 1 + (-1.92 - 1.92i)T + 47iT^{2} \) |
| 53 | \( 1 + (-5.52 - 3.01i)T + (28.6 + 44.5i)T^{2} \) |
| 59 | \( 1 + (1.80 - 6.13i)T + (-49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (4.00 - 6.23i)T + (-25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (0.554 + 7.74i)T + (-66.3 + 9.53i)T^{2} \) |
| 71 | \( 1 + (8.49 - 9.79i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (-1.19 - 0.891i)T + (20.5 + 70.0i)T^{2} \) |
| 79 | \( 1 + (-4.18 - 1.22i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (2.30 - 0.858i)T + (62.7 - 54.3i)T^{2} \) |
| 89 | \( 1 + (13.5 - 8.73i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (6.88 + 2.56i)T + (73.3 + 63.5i)T^{2} \) |
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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
−12.34196754816865481707256279583, −11.08068766203914800911964843398, −10.43725133639864920332782942514, −10.06866595221878612400848426756, −8.781856682827657805095144155742, −7.15441858654652228791882049163, −5.80603771508250232715751052125, −4.94633222101634699281149757573, −4.13741353345463680069635970413, −2.69450236929831441231070366623,
1.61589281837537741767551664149, 2.65261727957156702064049640880, 5.08441932483110020403816223653, 6.04960080643000926215603843748, 6.41867049335638704724727956980, 8.105437619345912611417800972186, 8.483727462954190857839812836686, 10.38865237721523407731422575771, 11.54058667514756689031412241022, 12.12884331004696866489141723769
