L(s) = 1 | + (1 − i)3-s + (−1 − i)7-s + i·9-s − 4i·11-s + (4 + 4i)13-s + (4 − 4i)17-s + 4·19-s − 2·21-s + (−5 + 5i)23-s + (4 + 4i)27-s − 2i·29-s − 8i·31-s + (−4 − 4i)33-s + 8·39-s − 4·41-s + ⋯ |
L(s) = 1 | + (0.577 − 0.577i)3-s + (−0.377 − 0.377i)7-s + 0.333i·9-s − 1.20i·11-s + (1.10 + 1.10i)13-s + (0.970 − 0.970i)17-s + 0.917·19-s − 0.436·21-s + (−1.04 + 1.04i)23-s + (0.769 + 0.769i)27-s − 0.371i·29-s − 1.43i·31-s + (−0.696 − 0.696i)33-s + 1.28·39-s − 0.624·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.135462563\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.135462563\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (-1 + i)T - 3iT^{2} \) |
| 7 | \( 1 + (1 + i)T + 7iT^{2} \) |
| 11 | \( 1 + 4iT - 11T^{2} \) |
| 13 | \( 1 + (-4 - 4i)T + 13iT^{2} \) |
| 17 | \( 1 + (-4 + 4i)T - 17iT^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + (5 - 5i)T - 23iT^{2} \) |
| 29 | \( 1 + 2iT - 29T^{2} \) |
| 31 | \( 1 + 8iT - 31T^{2} \) |
| 37 | \( 1 - 37iT^{2} \) |
| 41 | \( 1 + 4T + 41T^{2} \) |
| 43 | \( 1 + (-7 + 7i)T - 43iT^{2} \) |
| 47 | \( 1 + (3 + 3i)T + 47iT^{2} \) |
| 53 | \( 1 + (-4 - 4i)T + 53iT^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 + (3 + 3i)T + 67iT^{2} \) |
| 71 | \( 1 + 16iT - 71T^{2} \) |
| 73 | \( 1 + (4 + 4i)T + 73iT^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + (5 - 5i)T - 83iT^{2} \) |
| 89 | \( 1 - 10iT - 89T^{2} \) |
| 97 | \( 1 + (12 - 12i)T - 97iT^{2} \) |
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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
−9.233876344920512687445257134899, −8.377611628205301960469594209968, −7.71939737862736182821158411714, −7.03970656593713157818804602142, −6.05276863846924168607004081979, −5.33930526209935808973419937265, −3.90847410744804246205584291968, −3.29627641145198000554898142601, −2.09798820025580161533814140408, −0.902555752359531069755693787071,
1.28377457687275747464230304184, 2.78633052877549555790708846992, 3.51348826478406139020965012961, 4.33954070422947461421297813389, 5.48885816623283446822578987318, 6.21033591362749536666307052049, 7.18634752015541410659950370014, 8.251612302347543166735109945931, 8.621693339227168984376396147895, 9.753359233865044264880490250687