| L(s) = 1 | − 1.73i·3-s + 3.08·5-s − 2.99·9-s − 12.2i·11-s − 5.33i·15-s + 15.0·17-s − 24.5i·19-s − 8.51i·23-s − 15.4·25-s + 5.19i·27-s − 24.4·29-s − 3.75i·31-s − 21.2·33-s + 52.4·37-s − 39.0·41-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.616·5-s − 0.333·9-s − 1.11i·11-s − 0.355i·15-s + 0.887·17-s − 1.29i·19-s − 0.370i·23-s − 0.619·25-s + 0.192i·27-s − 0.844·29-s − 0.120i·31-s − 0.643·33-s + 1.41·37-s − 0.953·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.5i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.571613751\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.571613751\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 3.08T + 25T^{2} \) |
| 11 | \( 1 + 12.2iT - 121T^{2} \) |
| 13 | \( 1 + 169T^{2} \) |
| 17 | \( 1 - 15.0T + 289T^{2} \) |
| 19 | \( 1 + 24.5iT - 361T^{2} \) |
| 23 | \( 1 + 8.51iT - 529T^{2} \) |
| 29 | \( 1 + 24.4T + 841T^{2} \) |
| 31 | \( 1 + 3.75iT - 961T^{2} \) |
| 37 | \( 1 - 52.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + 39.0T + 1.68e3T^{2} \) |
| 43 | \( 1 - 27.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 21.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 11.5T + 2.80e3T^{2} \) |
| 59 | \( 1 - 42.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 29.5T + 3.72e3T^{2} \) |
| 67 | \( 1 + 73.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 40.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 42.4T + 5.32e3T^{2} \) |
| 79 | \( 1 + 23.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 83.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 148.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 186.T + 9.40e3T^{2} \) |
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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
−8.416964521298694222382113638947, −7.75413544030492747073795041588, −6.90792273437475664628811652482, −6.04319597279205545320888013024, −5.61003578996277685345063551069, −4.54776946015006314750175925608, −3.33517752338844294779687500726, −2.56303265906894222398659952307, −1.43330943212636808649736477247, −0.36890880631222701337140667921,
1.42919150021129011898515935730, 2.33759439015189382288009389951, 3.54741446404863325809019468598, 4.24688699426847388797794547727, 5.34643263772343069831866401664, 5.75781134747297443079710365070, 6.79588271761468923104935813118, 7.66878098961886354531453550743, 8.332155941281990028009573891526, 9.450021959040885793981044304035
