| L(s) = 1 | − 3-s + 5-s + 4.69i·7-s + 9-s − 1.75i·11-s − 4.69i·13-s − 15-s − 4.86·17-s + (3.21 − 2.93i)19-s − 4.69i·21-s + 4.44i·23-s + 25-s − 27-s − 7.14i·29-s − 6.21·31-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.77i·7-s + 0.333·9-s − 0.528i·11-s − 1.30i·13-s − 0.258·15-s − 1.17·17-s + (0.738 − 0.674i)19-s − 1.02i·21-s + 0.927i·23-s + 0.200·25-s − 0.192·27-s − 1.32i·29-s − 1.11·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.738 + 0.674i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.738 + 0.674i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2792407083\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2792407083\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + (-3.21 + 2.93i)T \) |
| good | 7 | \( 1 - 4.69iT - 7T^{2} \) |
| 11 | \( 1 + 1.75iT - 11T^{2} \) |
| 13 | \( 1 + 4.69iT - 13T^{2} \) |
| 17 | \( 1 + 4.86T + 17T^{2} \) |
| 23 | \( 1 - 4.44iT - 23T^{2} \) |
| 29 | \( 1 + 7.14iT - 29T^{2} \) |
| 31 | \( 1 + 6.21T + 31T^{2} \) |
| 37 | \( 1 - 6.20iT - 37T^{2} \) |
| 41 | \( 1 - 4.76iT - 41T^{2} \) |
| 43 | \( 1 + 6.20iT - 43T^{2} \) |
| 47 | \( 1 - 7.46iT - 47T^{2} \) |
| 53 | \( 1 + 4.44iT - 53T^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 + 15.3T + 61T^{2} \) |
| 67 | \( 1 + 11.7T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 + 9.79T + 73T^{2} \) |
| 79 | \( 1 - 1.35T + 79T^{2} \) |
| 83 | \( 1 - 10.8iT - 83T^{2} \) |
| 89 | \( 1 + 15.5iT - 89T^{2} \) |
| 97 | \( 1 + 1.67iT - 97T^{2} \) |
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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
−8.125458714009264880539720669773, −7.31445768740611719776820575609, −6.29031793419530173202698007998, −5.78707020693134202730954243841, −5.35740753280448780871491469822, −4.55091426725101299200509215674, −3.18700682160894034008658358353, −2.60433637488047036476121977686, −1.56172074261887356978357191787, −0.082790473617220268089771483705,
1.26384898693015541367715328521, 2.02774753734944774486006848890, 3.46599903625384082478339193224, 4.31879284719007739858769987447, 4.66691266193857000483089437985, 5.74185458174219248135720291363, 6.59750797559250530029878092630, 7.14122651256980704437795835573, 7.48537490301365114996291033739, 8.775329695760407177237927265780
