| L(s) = 1 | + (1.41 + 0.0247i)2-s + (−0.361 + 2.04i)3-s + (1.99 + 0.0700i)4-s + (−2.99 − 2.51i)5-s + (−0.561 + 2.88i)6-s + (−0.0108 + 0.00626i)7-s + (2.82 + 0.148i)8-s + (−1.25 − 0.455i)9-s + (−4.17 − 3.63i)10-s + (−3.15 − 1.82i)11-s + (−0.865 + 4.07i)12-s + (3.04 − 0.536i)13-s + (−0.0155 + 0.00859i)14-s + (6.24 − 5.23i)15-s + (3.99 + 0.279i)16-s + (−2.82 + 1.02i)17-s + ⋯ |
| L(s) = 1 | + (0.999 + 0.0175i)2-s + (−0.208 + 1.18i)3-s + (0.999 + 0.0350i)4-s + (−1.34 − 1.12i)5-s + (−0.229 + 1.17i)6-s + (−0.00410 + 0.00236i)7-s + (0.998 + 0.0525i)8-s + (−0.417 − 0.151i)9-s + (−1.32 − 1.14i)10-s + (−0.952 − 0.550i)11-s + (−0.249 + 1.17i)12-s + (0.844 − 0.148i)13-s + (−0.00414 + 0.00229i)14-s + (1.61 − 1.35i)15-s + (0.997 + 0.0699i)16-s + (−0.686 + 0.249i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.869 - 0.493i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.869 - 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.25360 + 0.330804i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.25360 + 0.330804i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.41 - 0.0247i)T \) |
| 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
−14.99722377771540882932615587447, −13.32974318829291344907142904085, −12.55599739829908430976605934580, −11.25687445856982925662820674877, −10.68453579365396204138292566508, −8.847858725047753759285535551553, −7.69925528532798444390143935989, −5.64877330549932787587582860885, −4.52029881054641722468705443005, −3.69448222703263475387544372482,
2.65147361011448055261926687937, 4.32394483908086047694959677991, 6.38203899090658716041133273751, 7.13069309390796452099414978487, 8.019909452794423725672543287390, 10.73433716422898530323304687172, 11.31873884597632677162527108023, 12.46622255531485063960825940068, 13.13970719093386759416579055838, 14.36795369473258059024557536760
