L(s) = 1 | + (−1 + i)3-s + (2 + i)5-s + (−1 − i)7-s + i·9-s − 4i·11-s + (−3 − 3i)13-s + (−3 + i)15-s + (−3 + 3i)17-s − 6·19-s + 2·21-s + (−3 + 3i)23-s + (3 + 4i)25-s + (−4 − 4i)27-s + 2i·29-s − 6i·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 − 1.20i·11-s + (−0.832 − 0.832i)13-s + (−0.774 + 0.258i)15-s + (−0.727 + 0.727i)17-s − 1.37·19-s + 0.436·21-s + (−0.625 + 0.625i)23-s + (0.600 + 0.800i)25-s + (−0.769 − 0.769i)27-s + 0.371i·29-s − 1.07i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 4iT - 11T^{2} \) |
| 13 | \( 1 + (3 + 3i)T + 13iT^{2} \) |
| 17 | \( 1 + (3 - 3i)T - 17iT^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 + (3 - 3i)T - 23iT^{2} \) |
| 29 | \( 1 - 2iT - 29T^{2} \) |
| 31 | \( 1 + 6iT - 31T^{2} \) |
| 37 | \( 1 + (3 - 3i)T - 37iT^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + (3 - 3i)T - 43iT^{2} \) |
| 47 | \( 1 + (-9 - 9i)T + 47iT^{2} \) |
| 53 | \( 1 + (5 + 5i)T + 53iT^{2} \) |
| 59 | \( 1 + 10T + 59T^{2} \) |
| 61 | \( 1 - 12T + 61T^{2} \) |
| 67 | \( 1 + (9 + 9i)T + 67iT^{2} \) |
| 71 | \( 1 - 6iT - 71T^{2} \) |
| 73 | \( 1 + (5 + 5i)T + 73iT^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + (-3 + 3i)T - 83iT^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + (7 - 7i)T - 97iT^{2} \) |
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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
−9.606299836933068250284171331592, −8.573553673398465119919932228310, −7.74162940427514495117161920140, −6.55790017168689678450898063968, −5.97872457856260773200563781721, −5.24034670808618061461712700234, −4.20491201335763035945327198834, −3.10793775980364593750376803825, −1.98380104940191069438811831423, 0,
1.76459039528730533636464057857, 2.46298842795261715058988739993, 4.23933687504990012576776971278, 4.99151588020140837812434783222, 5.98330834278771211460823296822, 6.74614447926115849110520562244, 7.14168588225264393809283129143, 8.632318355985944144310071390822, 9.199410844352709356249726533898