| L(s) = 1 | − 5.73i·3-s − 6.30i·5-s − 23.9·9-s − 5.41·11-s + 15.9i·13-s − 36.1·15-s − 20.4i·17-s + 13.5i·19-s − 4.70·23-s − 14.7·25-s + 85.7i·27-s + 1.76·29-s + 13.7i·31-s + 31.0i·33-s + 10.4·37-s + ⋯ |
| L(s) = 1 | − 1.91i·3-s − 1.26i·5-s − 2.66·9-s − 0.491·11-s + 1.22i·13-s − 2.41·15-s − 1.20i·17-s + 0.715i·19-s − 0.204·23-s − 0.589·25-s + 3.17i·27-s + 0.0608·29-s + 0.443i·31-s + 0.941i·33-s + 0.282·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.04580333337\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04580333337\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 5.73iT - 9T^{2} \) |
| 5 | \( 1 + 6.30iT - 25T^{2} \) |
| 11 | \( 1 + 5.41T + 121T^{2} \) |
| 13 | \( 1 - 15.9iT - 169T^{2} \) |
| 17 | \( 1 + 20.4iT - 289T^{2} \) |
| 19 | \( 1 - 13.5iT - 361T^{2} \) |
| 23 | \( 1 + 4.70T + 529T^{2} \) |
| 29 | \( 1 - 1.76T + 841T^{2} \) |
| 31 | \( 1 - 13.7iT - 961T^{2} \) |
| 37 | \( 1 - 10.4T + 1.36e3T^{2} \) |
| 41 | \( 1 - 11.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 49.1T + 1.84e3T^{2} \) |
| 47 | \( 1 + 10.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 32.1T + 2.80e3T^{2} \) |
| 59 | \( 1 - 66.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 31.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 98.5T + 4.48e3T^{2} \) |
| 71 | \( 1 + 61.7T + 5.04e3T^{2} \) |
| 73 | \( 1 - 18.0iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 30.0T + 6.24e3T^{2} \) |
| 83 | \( 1 + 63.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 137. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 131. iT - 9.40e3T^{2} \) |
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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
−8.600103456066974879413768792553, −7.60656265712493099154047600771, −7.10102358207896977091982774719, −6.17244814671183642407999638297, −5.40143594948070702032810427387, −4.50155664012945491966792969295, −2.95444922104442353293238537723, −1.88271995249614921805497621952, −1.13555320950782702843285079245, −0.01300852075436866857674574250,
2.54515291177328650305412048887, 3.22644057880130517929157422899, 3.94179634476416511244080658461, 4.96483405424449542853102911395, 5.72107365668179019214625284900, 6.50362839446396906251475783791, 7.77920790982300736736438640361, 8.451998864811265023935730235506, 9.390453013575598483661441465907, 10.20311156651857421349497711902
