L(s) = 1 | + (0.952 − 1.65i)2-s + 0.428·3-s + (−0.815 − 1.41i)4-s + (0.736 + 1.27i)5-s + (0.408 − 0.707i)6-s + (−2.62 − 0.311i)7-s + 0.702·8-s − 2.81·9-s + 2.80·10-s − 4.39·11-s + (−0.349 − 0.605i)12-s + (2.69 + 2.39i)13-s + (−3.01 + 4.03i)14-s + (0.315 + 0.546i)15-s + (2.30 − 3.98i)16-s + (0.601 + 1.04i)17-s + ⋯ |
L(s) = 1 | + (0.673 − 1.16i)2-s + 0.247·3-s + (−0.407 − 0.706i)4-s + (0.329 + 0.570i)5-s + (0.166 − 0.288i)6-s + (−0.993 − 0.117i)7-s + 0.248·8-s − 0.938·9-s + 0.887·10-s − 1.32·11-s + (−0.100 − 0.174i)12-s + (0.748 + 0.663i)13-s + (−0.806 + 1.07i)14-s + (0.0814 + 0.141i)15-s + (0.575 − 0.996i)16-s + (0.145 + 0.252i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.397 + 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.397 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.13238 - 0.743718i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.13238 - 0.743718i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (2.62 + 0.311i)T \) |
| 13 | \( 1 + (-2.69 - 2.39i)T \) |
good | 2 | \( 1 + (-0.952 + 1.65i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 - 0.428T + 3T^{2} \) |
| 5 | \( 1 + (-0.736 - 1.27i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + 4.39T + 11T^{2} \) |
| 17 | \( 1 + (-0.601 - 1.04i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 - 3.24T + 19T^{2} \) |
| 23 | \( 1 + (-2.21 + 3.84i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.0837 + 0.145i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (2.62 - 4.54i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.52 - 6.10i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.58 + 4.47i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.0113 - 0.0197i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (5.84 + 10.1i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.0708 + 0.122i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.67 - 4.62i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 - 4.13T + 67T^{2} \) |
| 71 | \( 1 + (-4.98 + 8.63i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (7.62 - 13.1i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.387 + 0.670i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 16.0T + 83T^{2} \) |
| 89 | \( 1 + (3.27 - 5.67i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.74 - 3.02i)T + (-48.5 - 84.0i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.64686853981771858663908290673, −12.93260300649249535544746771650, −11.79262756332152520627545143453, −10.72215415470714070835250365320, −10.03031337306835370899532674560, −8.520121825407119655694018246346, −6.84186418381644143599937427204, −5.34955969592241799584942325025, −3.49160821166836662000830581435, −2.57410502882543459336409725158,
3.24472151618467011588974359923, 5.28910117630892829738891696677, 5.85092102902327517521043883033, 7.37443527480713649089968317023, 8.447001911959619430498884380515, 9.702624843808053253425592211203, 11.10306832220830201724432802115, 12.93504544791923987631954489742, 13.23920140194018215877307343033, 14.29520029511420702331498968190