L(s) = 1 | + (−1.56 + 1.13i)2-s + (0.850 − 2.61i)4-s + (1.60 + 1.16i)5-s + (1.04 + 3.22i)8-s + (−0.809 + 0.587i)9-s − 3.84·10-s + (0.968 − 0.248i)11-s + (−0.866 + 0.629i)13-s + (−3.09 − 2.24i)16-s + (0.598 − 1.84i)18-s + (−0.115 − 0.356i)19-s + (4.41 − 3.21i)20-s + (−1.23 + 1.49i)22-s − 0.374·23-s + (0.907 + 2.79i)25-s + (0.641 − 1.97i)26-s + ⋯ |
L(s) = 1 | + (−1.56 + 1.13i)2-s + (0.850 − 2.61i)4-s + (1.60 + 1.16i)5-s + (1.04 + 3.22i)8-s + (−0.809 + 0.587i)9-s − 3.84·10-s + (0.968 − 0.248i)11-s + (−0.866 + 0.629i)13-s + (−3.09 − 2.24i)16-s + (0.598 − 1.84i)18-s + (−0.115 − 0.356i)19-s + (4.41 − 3.21i)20-s + (−1.23 + 1.49i)22-s − 0.374·23-s + (0.907 + 2.79i)25-s + (0.641 − 1.97i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 869 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.515 - 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 869 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.515 - 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5893024592\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5893024592\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (-0.968 + 0.248i)T \) |
| 79 | \( 1 + (0.809 - 0.587i)T \) |
good | 2 | \( 1 + (1.56 - 1.13i)T + (0.309 - 0.951i)T^{2} \) |
| 3 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 5 | \( 1 + (-1.60 - 1.16i)T + (0.309 + 0.951i)T^{2} \) |
| 7 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 13 | \( 1 + (0.866 - 0.629i)T + (0.309 - 0.951i)T^{2} \) |
| 17 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 19 | \( 1 + (0.115 + 0.356i)T + (-0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 + 0.374T + T^{2} \) |
| 29 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 31 | \( 1 + (-1.03 + 0.749i)T + (0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 53 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 67 | \( 1 + 0.851T + T^{2} \) |
| 71 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (-0.541 + 1.66i)T + (-0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 89 | \( 1 - 0.125T + T^{2} \) |
| 97 | \( 1 + (1.17 - 0.856i)T + (0.309 - 0.951i)T^{2} \) |
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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
−10.29249392824807761077650306787, −9.568846174802213670711065021162, −9.152869485997483538988976866054, −8.132421937399356696833545731043, −7.14058431713484979649916453237, −6.47443193987193210834511946828, −5.97211997926535628630405314611, −5.04713466749610905939924793121, −2.61432916058158973326059900528, −1.77787488811084348890061319681,
1.04080633391758185682198524141, 2.05045027916809333370138674364, 3.06314371282818272096183079348, 4.52508676160837799960325548454, 5.87262427641128419735989705087, 6.79686181198633638179306338355, 8.136917783688276932861761898162, 8.724882365508504639522742263586, 9.428552750621193949120419321273, 9.834075546300960520217777096878