L(s) = 1 | + i·2-s − 4-s + (−2.21 − 0.311i)5-s + 0.903i·7-s − i·8-s + (0.311 − 2.21i)10-s − 2·11-s + i·13-s − 0.903·14-s + 16-s − 5.33i·17-s + 1.52·19-s + (2.21 + 0.311i)20-s − 2i·22-s − 2.14i·23-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s + (−0.990 − 0.139i)5-s + 0.341i·7-s − 0.353i·8-s + (0.0983 − 0.700i)10-s − 0.603·11-s + 0.277i·13-s − 0.241·14-s + 0.250·16-s − 1.29i·17-s + 0.349·19-s + (0.495 + 0.0695i)20-s − 0.426i·22-s − 0.447i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.139i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 + 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.004124258\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.004124258\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.21 + 0.311i)T \) |
| 13 | \( 1 - iT \) |
good | 7 | \( 1 - 0.903iT - 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 17 | \( 1 + 5.33iT - 17T^{2} \) |
| 19 | \( 1 - 1.52T + 19T^{2} \) |
| 23 | \( 1 + 2.14iT - 23T^{2} \) |
| 29 | \( 1 - 5.65T + 29T^{2} \) |
| 31 | \( 1 - 2.42T + 31T^{2} \) |
| 37 | \( 1 + 3.24iT - 37T^{2} \) |
| 41 | \( 1 + 4.62T + 41T^{2} \) |
| 43 | \( 1 - 0.133iT - 43T^{2} \) |
| 47 | \( 1 + 5.80iT - 47T^{2} \) |
| 53 | \( 1 + 8.99iT - 53T^{2} \) |
| 59 | \( 1 - 9.05T + 59T^{2} \) |
| 61 | \( 1 - 12.6T + 61T^{2} \) |
| 67 | \( 1 + 10.8iT - 67T^{2} \) |
| 71 | \( 1 + 5.67T + 71T^{2} \) |
| 73 | \( 1 - 2.70iT - 73T^{2} \) |
| 79 | \( 1 - 6.91T + 79T^{2} \) |
| 83 | \( 1 - 10.2iT - 83T^{2} \) |
| 89 | \( 1 - 2.94T + 89T^{2} \) |
| 97 | \( 1 + 7.09iT - 97T^{2} \) |
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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
−9.608452953882772991106042257411, −8.670475079407333546019481617502, −8.151605367225589345543350519390, −7.23696031599669946412722405890, −6.66386115042538582682751995241, −5.36316478311575646574569978242, −4.79845409521555075451478553764, −3.72911011995714835084532732479, −2.60245615309758543206264223921, −0.54965333296788914856874267360,
1.05328662207236730763614273886, 2.63133787634348169894773941563, 3.58642940865768011749628786507, 4.35911795972726422442677900436, 5.34372594528953664753491852005, 6.51049192815861876604138342995, 7.56619492444439664256452554035, 8.186702975977321660614545348447, 8.913545291758918949051386030691, 10.24274264711553208130126993238