L(s) = 1 | + (−1 − 2i)5-s + 4i·13-s + 8i·17-s + (−3 + 4i)25-s + 10·29-s − 12i·37-s + 10·41-s + 7·49-s + 4i·53-s + 10·61-s + (8 − 4i)65-s + 16i·73-s + (16 − 8i)85-s − 10·89-s + 8i·97-s + ⋯ |
L(s) = 1 | + (−0.447 − 0.894i)5-s + 1.10i·13-s + 1.94i·17-s + (−0.600 + 0.800i)25-s + 1.85·29-s − 1.97i·37-s + 1.56·41-s + 49-s + 0.549i·53-s + 1.28·61-s + (0.992 − 0.496i)65-s + 1.87i·73-s + (1.73 − 0.867i)85-s − 1.05·89-s + 0.812i·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.431706806\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.431706806\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1 + 2i)T \) |
good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 - 8iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 10T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 12iT - 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 4iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 16iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 8iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.420844271631511952858485504146, −8.724196582595219545465983625642, −8.162169145789519053179197596571, −7.22451300132872997519292084108, −6.26634789343380574013748111604, −5.45167070235756735737882026857, −4.27324654952734499041190090775, −3.93107504365283413400261443332, −2.30702506626782030106039441031, −1.10903140412871219835651823945,
0.70081637769018333116178111448, 2.65571018849977082810313264801, 3.11797907156419186966667701453, 4.40214920062796692947090991854, 5.27114339318429747720415150274, 6.34110085891932730965192004051, 7.07682236961043110027976443929, 7.79941900119667763682971898495, 8.534964065312342776141551775522, 9.654155887792734734389719725953