L(s) = 1 | − i·3-s + (2 − i)5-s − 4i·7-s − 9-s − 2·11-s − i·13-s + (−1 − 2i)15-s + 4i·17-s + 2·19-s − 4·21-s − 6i·23-s + (3 − 4i)25-s + i·27-s + 2·29-s + 4·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + (0.894 − 0.447i)5-s − 1.51i·7-s − 0.333·9-s − 0.603·11-s − 0.277i·13-s + (−0.258 − 0.516i)15-s + 0.970i·17-s + 0.458·19-s − 0.872·21-s − 1.25i·23-s + (0.600 − 0.800i)25-s + 0.192i·27-s + 0.371·29-s + 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.737926617\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.737926617\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (-2 + i)T \) |
| 13 | \( 1 + iT \) |
good | 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 17 | \( 1 - 4iT - 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 + 10iT - 53T^{2} \) |
| 59 | \( 1 + 14T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 - 18T + 89T^{2} \) |
| 97 | \( 1 - 6iT - 97T^{2} \) |
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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
−8.203396142097353702658192561100, −7.73684288564244224274223851958, −6.72837551171062963927816350481, −6.31509713445386782909366190284, −5.29036757156116844128161869765, −4.58438088874882496699477259999, −3.59120485750435785431552244293, −2.49223479740000434515089261364, −1.46243885141435954861040462657, −0.52558357502861835195655003683,
1.63632091923718299215314129535, 2.75464441049776245883667430030, 3.05962880074096077920680276795, 4.56153144322455286771614073684, 5.39054375769800099634186200050, 5.72211124218642380475171809840, 6.62548920817660021423464808585, 7.51461354384254419417809268475, 8.483156927793049768175057895261, 9.158093684458620614520832841026