| 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
−11.14906847922311931928601933381, −10.43579626778532657147805579223, −8.736183255223080620201674924066, −8.423252027787823454970034099966, −7.43535707621193347353958107500, −6.49286641892705173524833259417, −5.90184090152618181790062485458, −3.71074893162472347673717190214, −2.86773499083883697092662610146, −1.97691614330000648598324634088,
1.10594409645189565792499003904, 3.23328933878498406490353995106, 4.36162074687148291429692865801, 4.87478833215905405012852537493, 6.29435078743970785211508936680, 7.06761773010742900835229855305, 8.833227215859727414560807848604, 9.110217782171113624711048483993, 10.02295072802836624911491379296, 10.83875684476117389469868737087
