| L(s) = 1 | − 8.04·5-s + 2·7-s − 21.7·11-s − 17.3i·13-s − 11.8i·17-s − 10.7i·19-s − 35.0i·23-s + 39.7·25-s − 11.7·29-s + 35.5·31-s − 16.0·35-s + 26i·37-s − 2.28i·41-s − 65.5i·43-s + 27.2i·47-s + ⋯ |
| L(s) = 1 | − 1.60·5-s + 0.285·7-s − 1.97·11-s − 1.33i·13-s − 0.697i·17-s − 0.566i·19-s − 1.52i·23-s + 1.59·25-s − 0.405·29-s + 1.14·31-s − 0.459·35-s + 0.702i·37-s − 0.0558i·41-s − 1.52i·43-s + 0.578i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.04918520579\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04918520579\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 8.04T + 25T^{2} \) |
| 7 | \( 1 - 2T + 49T^{2} \) |
| 11 | \( 1 + 21.7T + 121T^{2} \) |
| 13 | \( 1 + 17.3iT - 169T^{2} \) |
| 17 | \( 1 + 11.8iT - 289T^{2} \) |
| 19 | \( 1 + 10.7iT - 361T^{2} \) |
| 23 | \( 1 + 35.0iT - 529T^{2} \) |
| 29 | \( 1 + 11.7T + 841T^{2} \) |
| 31 | \( 1 - 35.5T + 961T^{2} \) |
| 37 | \( 1 - 26iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 2.28iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 65.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 27.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 49.5T + 2.80e3T^{2} \) |
| 59 | \( 1 + 73.5T + 3.48e3T^{2} \) |
| 61 | \( 1 - 7.52iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 65.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 84.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 84.5T + 5.32e3T^{2} \) |
| 79 | \( 1 + 75.5T + 6.24e3T^{2} \) |
| 83 | \( 1 + 48.2T + 6.88e3T^{2} \) |
| 89 | \( 1 + 146. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 106.T + 9.40e3T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.559411833093303625709966698767, −8.304367476684922989523905107492, −7.56166055362791279213360437895, −7.07546577824296819494313615739, −5.74516410919311644394703104263, −4.91699745721793190791361610763, −4.36899005444433437696791069110, −3.06968130717156429443019638449, −2.66425512358752201609064707301, −0.64698982760454149488193106192,
0.01953845524553485410730549640, 1.61560246524878289460492323633, 2.86843910603425727668359890234, 3.78108890089857300668706586432, 4.51660412410533174106371260318, 5.28713268174840765706327673309, 6.33413323639831641150668622081, 7.42626973948184548271231334672, 7.80655713661887450046232180187, 8.329403341534614514530396835230
