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
−9.753359233865044264880490250687, −8.621693339227168984376396147895, −8.251612302347543166735109945931, −7.18634752015541410659950370014, −6.21033591362749536666307052049, −5.48885816623283446822578987318, −4.33954070422947461421297813389, −3.51348826478406139020965012961, −2.78633052877549555790708846992, −1.28377457687275747464230304184,
0.902555752359531069755693787071, 2.09798820025580161533814140408, 3.29627641145198000554898142601, 3.90847410744804246205584291968, 5.33930526209935808973419937265, 6.05276863846924168607004081979, 7.03970656593713157818804602142, 7.71939737862736182821158411714, 8.377611628205301960469594209968, 9.233876344920512687445257134899