| L(s) = 1 | − 2.79i·2-s + 1.73·3-s − 3.79·4-s − 4.83i·6-s + (−5.63 + 4.15i)7-s − 0.578i·8-s + 2.99·9-s − 18.9·11-s − 6.56·12-s + 10.9·13-s + (11.6 + 15.7i)14-s − 16.7·16-s − 22.3·17-s − 8.37i·18-s − 19.6i·19-s + ⋯ |
| L(s) = 1 | − 1.39i·2-s + 0.577·3-s − 0.948·4-s − 0.805i·6-s + (−0.804 + 0.593i)7-s − 0.0723i·8-s + 0.333·9-s − 1.72·11-s − 0.547·12-s + 0.844·13-s + (0.829 + 1.12i)14-s − 1.04·16-s − 1.31·17-s − 0.465i·18-s − 1.03i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.171 - 0.985i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.171 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2800571775\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2800571775\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 1.73T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (5.63 - 4.15i)T \) |
| good | 2 | \( 1 + 2.79iT - 4T^{2} \) |
| 11 | \( 1 + 18.9T + 121T^{2} \) |
| 13 | \( 1 - 10.9T + 169T^{2} \) |
| 17 | \( 1 + 22.3T + 289T^{2} \) |
| 19 | \( 1 + 19.6iT - 361T^{2} \) |
| 23 | \( 1 - 31.9iT - 529T^{2} \) |
| 29 | \( 1 + 39.9T + 841T^{2} \) |
| 31 | \( 1 + 36.6iT - 961T^{2} \) |
| 37 | \( 1 - 8.94iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 37.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 18.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 49.3T + 2.20e3T^{2} \) |
| 53 | \( 1 + 49.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 35.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 63.4iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 21.3iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 36.2T + 5.04e3T^{2} \) |
| 73 | \( 1 - 6.66T + 5.32e3T^{2} \) |
| 79 | \( 1 - 16.2T + 6.24e3T^{2} \) |
| 83 | \( 1 - 36.7T + 6.88e3T^{2} \) |
| 89 | \( 1 + 88.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 133.T + 9.40e3T^{2} \) |
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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
−9.979625367955105994556848403950, −9.412430440819744686634945691549, −8.583607176954308430282194269302, −7.45492003254929533456077498553, −6.26181317845184779321658666665, −4.96868569727016804676288416398, −3.67043374041631392829644877801, −2.81986774628668238169267298760, −2.00028653825990356688059518337, −0.091962590291039263278730240329,
2.36501739696817116357049133080, 3.73963142643301500874716491704, 4.92924634484375879711967961475, 6.00867948393989498621273086927, 6.82349157478055796460467315489, 7.64546575364009848401744201834, 8.392562914466684120585999114083, 9.130975205328123587932912768876, 10.38849988405844732929841714320, 10.89607825644777226024643319498
