L(s) = 1 | − 1.83i·3-s − 1.83i·7-s − 0.364·9-s + 0.834·11-s + 2.19i·13-s + 2.56i·17-s + 19-s − 3.36·21-s − 0.635i·23-s − 4.83i·27-s + 9.62·29-s − 6.59·31-s − 1.53i·33-s − 5.23i·37-s + 4.03·39-s + ⋯ |
L(s) = 1 | − 1.05i·3-s − 0.693i·7-s − 0.121·9-s + 0.251·11-s + 0.609i·13-s + 0.621i·17-s + 0.229·19-s − 0.734·21-s − 0.132i·23-s − 0.930i·27-s + 1.78·29-s − 1.18·31-s − 0.266i·33-s − 0.860i·37-s + 0.645·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.861924536\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.861924536\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 + 1.83iT - 3T^{2} \) |
| 7 | \( 1 + 1.83iT - 7T^{2} \) |
| 11 | \( 1 - 0.834T + 11T^{2} \) |
| 13 | \( 1 - 2.19iT - 13T^{2} \) |
| 17 | \( 1 - 2.56iT - 17T^{2} \) |
| 23 | \( 1 + 0.635iT - 23T^{2} \) |
| 29 | \( 1 - 9.62T + 29T^{2} \) |
| 31 | \( 1 + 6.59T + 31T^{2} \) |
| 37 | \( 1 + 5.23iT - 37T^{2} \) |
| 41 | \( 1 - 4.43T + 41T^{2} \) |
| 43 | \( 1 + 7.06iT - 43T^{2} \) |
| 47 | \( 1 + 9.86iT - 47T^{2} \) |
| 53 | \( 1 - 0.668iT - 53T^{2} \) |
| 59 | \( 1 + 0.397T + 59T^{2} \) |
| 61 | \( 1 - 2.26T + 61T^{2} \) |
| 67 | \( 1 + 2.43iT - 67T^{2} \) |
| 71 | \( 1 - 4.12T + 71T^{2} \) |
| 73 | \( 1 + 9.49iT - 73T^{2} \) |
| 79 | \( 1 + 2.62T + 79T^{2} \) |
| 83 | \( 1 - 10.8iT - 83T^{2} \) |
| 89 | \( 1 - 5.97T + 89T^{2} \) |
| 97 | \( 1 - 10.5iT - 97T^{2} \) |
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
−8.153247589893984042804014398983, −7.35531945665832192930781275913, −6.90597384204016418864381360902, −6.29000741570215409440299144727, −5.36970284990587462464783735011, −4.30421526036297072065023350168, −3.69211282472426697328912494887, −2.40862489649484407571771670361, −1.58310955005510152360629385083, −0.61399921636274559247474804549,
1.16512632568380846914502097557, 2.60205291545495832822974439348, 3.27285231820678467931274711395, 4.25921993980672892949589834630, 4.91178129373782704811902515446, 5.59010999425794326914001971437, 6.41100390149538798418151000899, 7.31569831751891874675111182029, 8.131338473279670910477379655470, 8.908489495403596880836706590520