L(s) = 1 | − 4.46i·3-s + 8.47i·5-s − 10.8·9-s − 3.89·11-s − 19.1i·13-s + 37.7·15-s + 13.3i·17-s + 6.70i·19-s − 26·23-s − 46.7·25-s + 8.47i·27-s − 11.7·29-s − 36.5i·31-s + 17.3i·33-s − 32·37-s + ⋯ |
L(s) = 1 | − 1.48i·3-s + 1.69i·5-s − 1.21·9-s − 0.354·11-s − 1.47i·13-s + 2.51·15-s + 0.787i·17-s + 0.352i·19-s − 1.13·23-s − 1.87·25-s + 0.313i·27-s − 0.406·29-s − 1.18i·31-s + 0.527i·33-s − 0.864·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.912 - 0.409i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.912 - 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1439753835\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1439753835\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 4.46iT - 9T^{2} \) |
| 5 | \( 1 - 8.47iT - 25T^{2} \) |
| 11 | \( 1 + 3.89T + 121T^{2} \) |
| 13 | \( 1 + 19.1iT - 169T^{2} \) |
| 17 | \( 1 - 13.3iT - 289T^{2} \) |
| 19 | \( 1 - 6.70iT - 361T^{2} \) |
| 23 | \( 1 + 26T + 529T^{2} \) |
| 29 | \( 1 + 11.7T + 841T^{2} \) |
| 31 | \( 1 + 36.5iT - 961T^{2} \) |
| 37 | \( 1 + 32T + 1.36e3T^{2} \) |
| 41 | \( 1 - 20.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 79.2T + 1.84e3T^{2} \) |
| 47 | \( 1 + 14.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 13.7T + 2.80e3T^{2} \) |
| 59 | \( 1 - 8.45iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 31.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 31.3T + 4.48e3T^{2} \) |
| 71 | \( 1 + 95.5T + 5.04e3T^{2} \) |
| 73 | \( 1 + 18.3iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 79.7T + 6.24e3T^{2} \) |
| 83 | \( 1 - 141. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 116. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 137. iT - 9.40e3T^{2} \) |
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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
−9.941857801902668891636160640483, −8.177168682125685640456132832973, −7.86743441547196093776384364892, −7.00629350710781052723396960320, −6.26976623329021838241967253076, −5.61690355304032018470849197744, −3.66430112288343990616696500013, −2.73776716356675347233278417866, −1.80233436283480350807624596883, −0.04586815304997836175110735483,
1.74576473102836472670991367411, 3.48979934379371809347085710078, 4.55186182986605722164620814794, 4.82903224349747758037752896782, 5.79042914312167963498133443463, 7.19377856903608780158394154831, 8.564351030274062041848329430991, 8.896557699672286318586518970431, 9.671673964238272720080567779536, 10.28362728386167466646977786071