| L(s) = 1 | + (0.997 − 0.0713i)2-s + (−0.212 + 0.977i)3-s + (0.989 − 0.142i)4-s + (−2.14 − 0.627i)5-s + (−0.142 + 0.989i)6-s + (−0.236 + 0.433i)7-s + (0.977 − 0.212i)8-s + (−0.909 − 0.415i)9-s + (−2.18 − 0.472i)10-s + (−3.22 + 2.79i)11-s + (−0.0713 + 0.997i)12-s + (−4.47 + 2.44i)13-s + (−0.205 + 0.449i)14-s + (1.06 − 1.96i)15-s + (0.959 − 0.281i)16-s + (4.21 + 5.62i)17-s + ⋯ |
| L(s) = 1 | + (0.705 − 0.0504i)2-s + (−0.122 + 0.564i)3-s + (0.494 − 0.0711i)4-s + (−0.959 − 0.280i)5-s + (−0.0580 + 0.404i)6-s + (−0.0895 + 0.164i)7-s + (0.345 − 0.0751i)8-s + (−0.303 − 0.138i)9-s + (−0.691 − 0.149i)10-s + (−0.971 + 0.841i)11-s + (−0.0205 + 0.287i)12-s + (−1.24 + 0.677i)13-s + (−0.0548 + 0.120i)14-s + (0.276 − 0.507i)15-s + (0.239 − 0.0704i)16-s + (1.02 + 1.36i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.616 - 0.787i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.616 - 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.497813 + 1.02253i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.497813 + 1.02253i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.997 + 0.0713i)T \) |
| 3 | \( 1 + (0.212 - 0.977i)T \) |
| 5 | \( 1 + (2.14 + 0.627i)T \) |
| 23 | \( 1 + (1.95 + 4.37i)T \) |
| good | 7 | \( 1 + (0.236 - 0.433i)T + (-3.78 - 5.88i)T^{2} \) |
| 11 | \( 1 + (3.22 - 2.79i)T + (1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (4.47 - 2.44i)T + (7.02 - 10.9i)T^{2} \) |
| 17 | \( 1 + (-4.21 - 5.62i)T + (-4.78 + 16.3i)T^{2} \) |
| 19 | \( 1 + (-1.01 - 7.03i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (4.61 + 0.662i)T + (27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (-0.0976 - 0.0627i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (1.46 + 3.92i)T + (-27.9 + 24.2i)T^{2} \) |
| 41 | \( 1 + (3.92 + 8.58i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-7.47 - 1.62i)T + (39.1 + 17.8i)T^{2} \) |
| 47 | \( 1 + (-0.867 - 0.867i)T + 47iT^{2} \) |
| 53 | \( 1 + (8.31 + 4.53i)T + (28.6 + 44.5i)T^{2} \) |
| 59 | \( 1 + (1.73 - 5.92i)T + (-49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (2.51 - 3.90i)T + (-25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (0.102 + 1.43i)T + (-66.3 + 9.53i)T^{2} \) |
| 71 | \( 1 + (0.855 - 0.987i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (4.89 + 3.66i)T + (20.5 + 70.0i)T^{2} \) |
| 79 | \( 1 + (-6.31 - 1.85i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (-15.1 + 5.64i)T + (62.7 - 54.3i)T^{2} \) |
| 89 | \( 1 + (-7.84 + 5.04i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (-17.2 - 6.44i)T + (73.3 + 63.5i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.61869483495056304466125745593, −10.23882085168953895472853752645, −9.141960889840050453957852626896, −7.87382721176551465866959875727, −7.48073324625097312572852589603, −6.05987001664658300802666596957, −5.16274595503663569958858076922, −4.29658698580139966342621818682, −3.54039946891268818564119044691, −2.10447134414527041192878909611,
0.45793942093144317941570804996, 2.72115373551113556261753427557, 3.28149643352718807632402784219, 4.86814703283529307592213831878, 5.42129343567150517118465904253, 6.74405996333751561454442449831, 7.62714145104520273790696705748, 7.83289409290319592429479088573, 9.324220613369593557767547896253, 10.42997058356217631223297034673
