| L(s) = 1 | + (0.182 + 0.228i)2-s + (0.878 + 2.23i)3-s + (0.426 − 1.86i)4-s + (0.135 − 1.80i)5-s + (−0.350 + 0.607i)6-s + (−1.86 − 3.23i)7-s + (1.02 − 0.495i)8-s + (−2.03 + 1.88i)9-s + (0.436 − 0.297i)10-s + (−0.222 − 0.974i)11-s + (4.55 − 0.685i)12-s + (5.89 + 4.01i)13-s + (0.398 − 1.01i)14-s + (4.15 − 1.28i)15-s + (−3.14 − 1.51i)16-s + (−0.313 − 4.18i)17-s + ⋯ |
| L(s) = 1 | + (0.128 + 0.161i)2-s + (0.506 + 1.29i)3-s + (0.213 − 0.933i)4-s + (0.0605 − 0.807i)5-s + (−0.143 + 0.248i)6-s + (−0.706 − 1.22i)7-s + (0.364 − 0.175i)8-s + (−0.678 + 0.629i)9-s + (0.138 − 0.0942i)10-s + (−0.0670 − 0.293i)11-s + (1.31 − 0.197i)12-s + (1.63 + 1.11i)13-s + (0.106 − 0.271i)14-s + (1.07 − 0.331i)15-s + (−0.787 − 0.379i)16-s + (−0.0760 − 1.01i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 473 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 + 0.389i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 473 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.921 + 0.389i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.76252 - 0.357292i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.76252 - 0.357292i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (0.222 + 0.974i)T \) |
| 43 | \( 1 + (2.56 - 6.03i)T \) |
| good | 2 | \( 1 + (-0.182 - 0.228i)T + (-0.445 + 1.94i)T^{2} \) |
| 3 | \( 1 + (-0.878 - 2.23i)T + (-2.19 + 2.04i)T^{2} \) |
| 5 | \( 1 + (-0.135 + 1.80i)T + (-4.94 - 0.745i)T^{2} \) |
| 7 | \( 1 + (1.86 + 3.23i)T + (-3.5 + 6.06i)T^{2} \) |
| 13 | \( 1 + (-5.89 - 4.01i)T + (4.74 + 12.1i)T^{2} \) |
| 17 | \( 1 + (0.313 + 4.18i)T + (-16.8 + 2.53i)T^{2} \) |
| 19 | \( 1 + (4.45 + 4.12i)T + (1.41 + 18.9i)T^{2} \) |
| 23 | \( 1 + (-4.81 - 1.48i)T + (19.0 + 12.9i)T^{2} \) |
| 29 | \( 1 + (1.15 - 2.94i)T + (-21.2 - 19.7i)T^{2} \) |
| 31 | \( 1 + (0.439 - 0.0662i)T + (29.6 - 9.13i)T^{2} \) |
| 37 | \( 1 + (2.71 - 4.70i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.46 - 5.60i)T + (-9.12 + 39.9i)T^{2} \) |
| 47 | \( 1 + (-0.978 + 4.28i)T + (-42.3 - 20.3i)T^{2} \) |
| 53 | \( 1 + (4.29 - 2.92i)T + (19.3 - 49.3i)T^{2} \) |
| 59 | \( 1 + (-12.8 - 6.18i)T + (36.7 + 46.1i)T^{2} \) |
| 61 | \( 1 + (13.4 + 2.02i)T + (58.2 + 17.9i)T^{2} \) |
| 67 | \( 1 + (2.56 + 2.37i)T + (5.00 + 66.8i)T^{2} \) |
| 71 | \( 1 + (3.39 - 1.04i)T + (58.6 - 39.9i)T^{2} \) |
| 73 | \( 1 + (-4.18 - 2.85i)T + (26.6 + 67.9i)T^{2} \) |
| 79 | \( 1 + (4.91 + 8.50i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (4.32 + 11.0i)T + (-60.8 + 56.4i)T^{2} \) |
| 89 | \( 1 + (-1.45 - 3.71i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + (-2.73 - 11.9i)T + (-87.3 + 42.0i)T^{2} \) |
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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
−10.83875684476117389469868737087, −10.02295072802836624911491379296, −9.110217782171113624711048483993, −8.833227215859727414560807848604, −7.06761773010742900835229855305, −6.29435078743970785211508936680, −4.87478833215905405012852537493, −4.36162074687148291429692865801, −3.23328933878498406490353995106, −1.10594409645189565792499003904,
1.97691614330000648598324634088, 2.86773499083883697092662610146, 3.71074893162472347673717190214, 5.90184090152618181790062485458, 6.49286641892705173524833259417, 7.43535707621193347353958107500, 8.423252027787823454970034099966, 8.736183255223080620201674924066, 10.43579626778532657147805579223, 11.14906847922311931928601933381
