| L(s) = 1 | + (−1 + i)3-s + (−2 − i)5-s + (−1 + i)7-s + i·9-s + 6·11-s + (−1 − i)13-s + (3 − i)15-s + (1 + i)17-s − 4i·19-s − 2i·21-s + (5 + 5i)23-s + (3 + 4i)25-s + (−4 − 4i)27-s − 8·29-s − 2i·31-s + ⋯ |
| L(s) = 1 | + (−0.577 + 0.577i)3-s + (−0.894 − 0.447i)5-s + (−0.377 + 0.377i)7-s + 0.333i·9-s + 1.80·11-s + (−0.277 − 0.277i)13-s + (0.774 − 0.258i)15-s + (0.242 + 0.242i)17-s − 0.917i·19-s − 0.436i·21-s + (1.04 + 1.04i)23-s + (0.600 + 0.800i)25-s + (−0.769 − 0.769i)27-s − 1.48·29-s − 0.359i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.767 - 0.640i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.767 - 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6653750659\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6653750659\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (2 + i)T \) |
| good | 3 | \( 1 + (1 - i)T - 3iT^{2} \) |
| 7 | \( 1 + (1 - i)T - 7iT^{2} \) |
| 11 | \( 1 - 6T + 11T^{2} \) |
| 13 | \( 1 + (1 + i)T + 13iT^{2} \) |
| 17 | \( 1 + (-1 - i)T + 17iT^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + (-5 - 5i)T + 23iT^{2} \) |
| 29 | \( 1 + 8T + 29T^{2} \) |
| 31 | \( 1 + 2iT - 31T^{2} \) |
| 37 | \( 1 + (5 - 5i)T - 37iT^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + (3 - 3i)T - 43iT^{2} \) |
| 47 | \( 1 + (7 - 7i)T - 47iT^{2} \) |
| 53 | \( 1 + (-1 - i)T + 53iT^{2} \) |
| 59 | \( 1 - 4iT - 59T^{2} \) |
| 61 | \( 1 - 2iT - 61T^{2} \) |
| 67 | \( 1 + (-7 - 7i)T + 67iT^{2} \) |
| 71 | \( 1 + 6iT - 71T^{2} \) |
| 73 | \( 1 + (9 - 9i)T - 73iT^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + (5 - 5i)T - 83iT^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + (3 + 3i)T + 97iT^{2} \) |
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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
−9.807735354794905581530592079613, −9.272072445653562646300314072668, −8.511109683962489576416988926912, −7.48038961464668760562434443057, −6.71416129302840148502117855105, −5.64444594093886384651758204454, −4.87166318245351595972247983051, −4.03753841264483963138599175312, −3.17707480947023233591466577047, −1.38527365421148303085833712471,
0.33044886942901098316539443010, 1.65171404125528601385392377384, 3.48688916936832019797727492340, 3.84986427928083685601883105354, 5.16737897252026642918622899044, 6.44056160427996727909260292567, 6.77957501910312377553814405254, 7.40452724078270674606199346892, 8.568888306870634378109142751078, 9.303216815984415969681090537131
