| L(s) = 1 | + (−0.415 + 0.909i)2-s + (−0.654 − 0.755i)4-s + (−0.767 + 0.492i)5-s + (−0.601 + 4.18i)7-s + (0.959 − 0.281i)8-s + (−0.129 − 0.902i)10-s + (−0.630 − 1.38i)11-s + (−0.0694 − 0.483i)13-s + (−3.55 − 2.28i)14-s + (−0.142 + 0.989i)16-s + (−2.10 + 2.43i)17-s + (−3.48 − 4.01i)19-s + (0.874 + 0.256i)20-s + 1.51·22-s + (−2.89 + 3.82i)23-s + ⋯ |
| L(s) = 1 | + (−0.293 + 0.643i)2-s + (−0.327 − 0.377i)4-s + (−0.343 + 0.220i)5-s + (−0.227 + 1.58i)7-s + (0.339 − 0.0996i)8-s + (−0.0410 − 0.285i)10-s + (−0.190 − 0.416i)11-s + (−0.0192 − 0.134i)13-s + (−0.950 − 0.611i)14-s + (−0.0355 + 0.247i)16-s + (−0.511 + 0.590i)17-s + (−0.799 − 0.922i)19-s + (0.195 + 0.0574i)20-s + 0.323·22-s + (−0.603 + 0.797i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.193i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 - 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0617278 + 0.632281i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0617278 + 0.632281i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.415 - 0.909i)T \) |
| 3 | \( 1 \) |
| 23 | \( 1 + (2.89 - 3.82i)T \) |
| good | 5 | \( 1 + (0.767 - 0.492i)T + (2.07 - 4.54i)T^{2} \) |
| 7 | \( 1 + (0.601 - 4.18i)T + (-6.71 - 1.97i)T^{2} \) |
| 11 | \( 1 + (0.630 + 1.38i)T + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (0.0694 + 0.483i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (2.10 - 2.43i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (3.48 + 4.01i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (5.24 - 6.05i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (7.89 - 2.31i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (-6.47 - 4.16i)T + (15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (-7.73 + 4.97i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (-4.30 - 1.26i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 - 0.273T + 47T^{2} \) |
| 53 | \( 1 + (-1.04 + 7.26i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (0.161 + 1.12i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (-13.8 + 4.07i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (-0.851 + 1.86i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (4.40 - 9.65i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (0.420 + 0.485i)T + (-10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (-0.230 - 1.60i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (-10.0 - 6.47i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (-8.97 - 2.63i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (3.83 - 2.46i)T + (40.2 - 88.2i)T^{2} \) |
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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
−11.45608554856078125332639734109, −10.85025282088379282582726946986, −9.447606135670753696064316324976, −8.916585608571363960842469177874, −8.047068523220256305713489203315, −6.98356791561540002018023635664, −5.93165951523204105635956438110, −5.25441312123679541049134594434, −3.68878940439734557335642693401, −2.21845624107307297516789551680,
0.43470611364481047247838805771, 2.22744203663515981722726191031, 3.99104695589624440199703319354, 4.32032646210321766958556713948, 6.10316706938246705441469050086, 7.39168277697233991393680176083, 7.911197464104255978229804793546, 9.185710634198043012787936054394, 10.04382487664344936428013626743, 10.73429731647771012023045929018
