L(s) = 1 | + (−1.06 − 0.927i)2-s + (0.361 + 2.04i)3-s + (0.278 + 1.98i)4-s + (−2.99 + 2.51i)5-s + (1.51 − 2.52i)6-s + (0.0108 + 0.00626i)7-s + (1.54 − 2.37i)8-s + (−1.25 + 0.455i)9-s + (5.53 + 0.0969i)10-s + (3.15 − 1.82i)11-s + (−3.95 + 1.28i)12-s + (3.04 + 0.536i)13-s + (−0.00576 − 0.0167i)14-s + (−6.24 − 5.23i)15-s + (−3.84 + 1.10i)16-s + (−2.82 − 1.02i)17-s + ⋯ |
L(s) = 1 | + (−0.754 − 0.656i)2-s + (0.208 + 1.18i)3-s + (0.139 + 0.990i)4-s + (−1.34 + 1.12i)5-s + (0.618 − 1.03i)6-s + (0.00410 + 0.00236i)7-s + (0.544 − 0.838i)8-s + (−0.417 + 0.151i)9-s + (1.75 + 0.0306i)10-s + (0.952 − 0.550i)11-s + (−1.14 + 0.371i)12-s + (0.844 + 0.148i)13-s + (−0.00154 − 0.00447i)14-s + (−1.61 − 1.35i)15-s + (−0.961 + 0.275i)16-s + (−0.686 − 0.249i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.400 - 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.400 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.515613 + 0.337545i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.515613 + 0.337545i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.06 + 0.927i)T \) |
| 19 | \( 1 + (-3.96 - 1.81i)T \) |
good | 3 | \( 1 + (-0.361 - 2.04i)T + (-2.81 + 1.02i)T^{2} \) |
| 5 | \( 1 + (2.99 - 2.51i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (-0.0108 - 0.00626i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-3.15 + 1.82i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.04 - 0.536i)T + (12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (2.82 + 1.02i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (2.50 - 2.97i)T + (-3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (0.126 + 0.347i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-1.29 + 2.24i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 2.87iT - 37T^{2} \) |
| 41 | \( 1 + (-7.88 + 1.39i)T + (38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (5.11 + 6.09i)T + (-7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (2.68 + 7.37i)T + (-36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (0.993 - 1.18i)T + (-9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (-3.93 - 1.43i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (2.45 + 2.05i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (8.76 - 3.18i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-9.25 + 7.76i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (-0.801 - 4.54i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (0.148 + 0.844i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-4.31 - 2.49i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-4.86 - 0.858i)T + (83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (1.35 - 3.70i)T + (-74.3 - 62.3i)T^{2} \) |
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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
−15.03625629868113861527474384168, −13.77222458349178731952327401992, −11.84152734156100456860501445012, −11.29284837721063440855197128332, −10.38209665121112581599987994380, −9.254325996194497550746373445431, −8.127624535764367727711468221547, −6.81657782272243952902492033605, −4.00901212134792904127181113476, −3.38973916220022140047962941495,
1.20184909762105019397197168795, 4.50313483586985032137055263256, 6.43165580582090097832045034354, 7.55516982146695724670579867562, 8.319651583787419691160732358305, 9.277241709142559674058066759900, 11.20310634757337556427386105587, 12.18804405842626148007318414498, 13.17627971028029015959518870003, 14.42813878355236440291819494024