| 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
−14.42813878355236440291819494024, −13.17627971028029015959518870003, −12.18804405842626148007318414498, −11.20310634757337556427386105587, −9.277241709142559674058066759900, −8.319651583787419691160732358305, −7.55516982146695724670579867562, −6.43165580582090097832045034354, −4.50313483586985032137055263256, −1.20184909762105019397197168795,
3.38973916220022140047962941495, 4.00901212134792904127181113476, 6.81657782272243952902492033605, 8.127624535764367727711468221547, 9.254325996194497550746373445431, 10.38209665121112581599987994380, 11.29284837721063440855197128332, 11.84152734156100456860501445012, 13.77222458349178731952327401992, 15.03625629868113861527474384168
