| L(s) = 1 | + (−0.545 − 0.160i)2-s + (−0.654 + 0.755i)3-s + (−1.41 − 0.906i)4-s + (−0.339 + 2.35i)5-s + (0.478 − 0.307i)6-s + (−0.415 − 0.909i)7-s + (1.36 + 1.58i)8-s + (−0.142 − 0.989i)9-s + (0.563 − 1.23i)10-s + (−4.74 + 1.39i)11-s + (1.60 − 0.472i)12-s + (2.17 − 4.75i)13-s + (0.0809 + 0.562i)14-s + (−1.56 − 1.80i)15-s + (0.899 + 1.96i)16-s + (2.04 − 1.31i)17-s + ⋯ |
| L(s) = 1 | + (−0.385 − 0.113i)2-s + (−0.378 + 0.436i)3-s + (−0.705 − 0.453i)4-s + (−0.151 + 1.05i)5-s + (0.195 − 0.125i)6-s + (−0.157 − 0.343i)7-s + (0.484 + 0.558i)8-s + (−0.0474 − 0.329i)9-s + (0.178 − 0.389i)10-s + (−1.43 + 0.420i)11-s + (0.464 − 0.136i)12-s + (0.602 − 1.31i)13-s + (0.0216 + 0.150i)14-s + (−0.403 − 0.465i)15-s + (0.224 + 0.492i)16-s + (0.495 − 0.318i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0182 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0182 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.324940 - 0.319062i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.324940 - 0.319062i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.654 - 0.755i)T \) |
| 7 | \( 1 + (0.415 + 0.909i)T \) |
| 23 | \( 1 + (-2.01 + 4.35i)T \) |
| good | 2 | \( 1 + (0.545 + 0.160i)T + (1.68 + 1.08i)T^{2} \) |
| 5 | \( 1 + (0.339 - 2.35i)T + (-4.79 - 1.40i)T^{2} \) |
| 11 | \( 1 + (4.74 - 1.39i)T + (9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (-2.17 + 4.75i)T + (-8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (-2.04 + 1.31i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (0.689 + 0.442i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (-1.50 + 0.967i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (6.14 + 7.09i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (1.31 + 9.16i)T + (-35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (-1.17 + 8.14i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (1.32 - 1.52i)T + (-6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + 1.45T + 47T^{2} \) |
| 53 | \( 1 + (-1.66 - 3.64i)T + (-34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (3.61 - 7.90i)T + (-38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (2.09 + 2.41i)T + (-8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (-9.40 - 2.76i)T + (56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (-2.93 - 0.861i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (10.1 + 6.49i)T + (30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (0.292 - 0.640i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (-0.992 - 6.90i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (-2.45 + 2.82i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (1.89 - 13.1i)T + (-93.0 - 27.3i)T^{2} \) |
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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
−10.59142849249938257533379369280, −10.22090992076453917057978948257, −9.173621560056459786027187519085, −8.022588551254545208957192239247, −7.27021033451911740494487304857, −5.87944522172112192545027540622, −5.16973177061108843768089921431, −3.90696043899553641413157914646, −2.65930878295480581779629464484, −0.36485328181033486676397298802,
1.36359900701967397174702071795, 3.34736933399401678719305491739, 4.72244683966334951064556920135, 5.38998757829600417576096503186, 6.71075773980331055293066679355, 7.86339162093024890310753955572, 8.490145150665688362011979026029, 9.162368421671533441867162656016, 10.18602143867842706485131637500, 11.30031926649207852770063356902
