L(s) = 1 | + (−1 + i)2-s + (−1.22 − 1.22i)3-s − 2i·4-s + 2.44·6-s + (−1.87 + 1.87i)7-s + (2 + 2i)8-s + 2.99i·9-s − 19.6·11-s + (−2.44 + 2.44i)12-s + (−0.747 − 0.747i)13-s − 3.74i·14-s − 4·16-s + (20.6 − 20.6i)17-s + (−2.99 − 2.99i)18-s + 13.7i·19-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.5i)2-s + (−0.408 − 0.408i)3-s − 0.5i·4-s + 0.408·6-s + (−0.267 + 0.267i)7-s + (0.250 + 0.250i)8-s + 0.333i·9-s − 1.78·11-s + (−0.204 + 0.204i)12-s + (−0.0574 − 0.0574i)13-s − 0.267i·14-s − 0.250·16-s + (1.21 − 1.21i)17-s + (−0.166 − 0.166i)18-s + 0.721i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 - 0.326i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.945 - 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8347537758\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8347537758\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 - i)T \) |
| 3 | \( 1 + (1.22 + 1.22i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1.87 - 1.87i)T \) |
good | 11 | \( 1 + 19.6T + 121T^{2} \) |
| 13 | \( 1 + (0.747 + 0.747i)T + 169iT^{2} \) |
| 17 | \( 1 + (-20.6 + 20.6i)T - 289iT^{2} \) |
| 19 | \( 1 - 13.7iT - 361T^{2} \) |
| 23 | \( 1 + (15.2 + 15.2i)T + 529iT^{2} \) |
| 29 | \( 1 - 52.0iT - 841T^{2} \) |
| 31 | \( 1 - 6.68T + 961T^{2} \) |
| 37 | \( 1 + (34.0 - 34.0i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 2.98T + 1.68e3T^{2} \) |
| 43 | \( 1 + (16.2 + 16.2i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-62.2 + 62.2i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-21.3 - 21.3i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + 95.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 31.9T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-41.4 + 41.4i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 - 108.T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-79.9 - 79.9i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 - 87.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (17.2 + 17.2i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 - 60.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-49.0 + 49.0i)T - 9.40e3iT^{2} \) |
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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
−9.992207174624559510709151378885, −8.736559352042101676419860903339, −7.998856453944803489674115798741, −7.33935668596094521624932510392, −6.50797974638386518499978263801, −5.30603783408655017864444867474, −5.18928713587367379104531861574, −3.30045904930697423820925178136, −2.16242485265301082084770871683, −0.62123331105063167275318555110,
0.57516907752235252399628029504, 2.19100286421403794814699162810, 3.29660428050365296534939068369, 4.27217671037214156789467702251, 5.39229227783854526276414583039, 6.14144419882289943154566577352, 7.53072343486918848006928124754, 7.936882474551648575634900001580, 9.011495837781891975369560532606, 10.03826607563062120529471232105