| L(s) = 1 | − 1.41·2-s − 1.73·3-s + 2.00·4-s + 2.23i·5-s + 2.44·6-s + 8.87i·7-s − 2.82·8-s + 2.99·9-s − 3.16i·10-s − 6.26i·11-s − 3.46·12-s + 22.5·13-s − 12.5i·14-s − 3.87i·15-s + 4.00·16-s − 9.43i·17-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.500·4-s + 0.447i·5-s + 0.408·6-s + 1.26i·7-s − 0.353·8-s + 0.333·9-s − 0.316i·10-s − 0.569i·11-s − 0.288·12-s + 1.73·13-s − 0.896i·14-s − 0.258i·15-s + 0.250·16-s − 0.555i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.141 - 0.989i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.141 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.054793336\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.054793336\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 + 1.73T \) |
| 5 | \( 1 - 2.23iT \) |
| 23 | \( 1 + (-22.7 - 3.26i)T \) |
| good | 7 | \( 1 - 8.87iT - 49T^{2} \) |
| 11 | \( 1 + 6.26iT - 121T^{2} \) |
| 13 | \( 1 - 22.5T + 169T^{2} \) |
| 17 | \( 1 + 9.43iT - 289T^{2} \) |
| 19 | \( 1 - 10.5iT - 361T^{2} \) |
| 29 | \( 1 + 7.04T + 841T^{2} \) |
| 31 | \( 1 + 18.9T + 961T^{2} \) |
| 37 | \( 1 + 39.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 2.96T + 1.68e3T^{2} \) |
| 43 | \( 1 - 73.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 64.8T + 2.20e3T^{2} \) |
| 53 | \( 1 + 14.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 19.3T + 3.48e3T^{2} \) |
| 61 | \( 1 - 55.1iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 41.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 33.4T + 5.04e3T^{2} \) |
| 73 | \( 1 - 40.7T + 5.32e3T^{2} \) |
| 79 | \( 1 - 49.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 36.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 129. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 101. iT - 9.40e3T^{2} \) |
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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
−10.68707247246859718947471797425, −9.434836261687970426110405777881, −8.856632086283408985142542614987, −7.999900783692044794666482343102, −6.87853807160162589018366454986, −5.98131901417909254538632411922, −5.46858262310932217207272728977, −3.74566205697404643491905295939, −2.59279503845630645533702098544, −1.15533518522722574332720516084,
0.63214457974345811345916993812, 1.59647551000355018259375791009, 3.54739735843656693996799799103, 4.48898961392553162968074976499, 5.70854811483569625182170997968, 6.72490727855329873677083823326, 7.37144705748309816343362648210, 8.437152602917169515553322260058, 9.173075599862444118546025091559, 10.26966752240322655800640516216
