L(s) = 1 | + (−0.573 − 0.819i)2-s + (−0.342 + 0.939i)4-s + (−3.25 − 0.284i)5-s + (−0.761 + 0.638i)7-s + (0.965 − 0.258i)8-s + (1.63 + 2.83i)10-s + (2.46 − 4.26i)11-s + (−3.00 + 1.40i)13-s + (0.960 + 0.257i)14-s + (−0.766 − 0.642i)16-s + (−0.884 − 0.412i)17-s + (6.51 + 4.56i)19-s + (1.38 − 2.96i)20-s + (−4.91 + 0.429i)22-s + (−1.73 + 6.46i)23-s + ⋯ |
L(s) = 1 | + (−0.405 − 0.579i)2-s + (−0.171 + 0.469i)4-s + (−1.45 − 0.127i)5-s + (−0.287 + 0.241i)7-s + (0.341 − 0.0915i)8-s + (0.516 + 0.895i)10-s + (0.743 − 1.28i)11-s + (−0.833 + 0.388i)13-s + (0.256 + 0.0687i)14-s + (−0.191 − 0.160i)16-s + (−0.214 − 0.100i)17-s + (1.49 + 1.04i)19-s + (0.308 − 0.662i)20-s + (−1.04 + 0.0916i)22-s + (−0.361 + 1.34i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.903 - 0.429i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.903 - 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.698686 + 0.157674i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.698686 + 0.157674i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.573 + 0.819i)T \) |
| 3 | \( 1 \) |
| 37 | \( 1 + (-2.37 + 5.60i)T \) |
good | 5 | \( 1 + (3.25 + 0.284i)T + (4.92 + 0.868i)T^{2} \) |
| 7 | \( 1 + (0.761 - 0.638i)T + (1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (-2.46 + 4.26i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (3.00 - 1.40i)T + (8.35 - 9.95i)T^{2} \) |
| 17 | \( 1 + (0.884 + 0.412i)T + (10.9 + 13.0i)T^{2} \) |
| 19 | \( 1 + (-6.51 - 4.56i)T + (6.49 + 17.8i)T^{2} \) |
| 23 | \( 1 + (1.73 - 6.46i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (-1.24 - 4.64i)T + (-25.1 + 14.5i)T^{2} \) |
| 31 | \( 1 + (-7.57 - 7.57i)T + 31iT^{2} \) |
| 41 | \( 1 + (-7.43 - 2.70i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (1.60 - 1.60i)T - 43iT^{2} \) |
| 47 | \( 1 + (-3.88 + 2.24i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (7.06 - 8.42i)T + (-9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (0.720 + 8.23i)T + (-58.1 + 10.2i)T^{2} \) |
| 61 | \( 1 + (3.71 + 7.97i)T + (-39.2 + 46.7i)T^{2} \) |
| 67 | \( 1 + (1.59 + 1.89i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (5.07 - 0.894i)T + (66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + 5.03iT - 73T^{2} \) |
| 79 | \( 1 + (-0.278 + 3.18i)T + (-77.7 - 13.7i)T^{2} \) |
| 83 | \( 1 + (-3.09 - 8.50i)T + (-63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (5.91 - 0.517i)T + (87.6 - 15.4i)T^{2} \) |
| 97 | \( 1 + (-11.4 - 3.07i)T + (84.0 + 48.5i)T^{2} \) |
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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
−10.75488841049895515156986826504, −9.577155114273327096188387629573, −8.971504096264490286735924156494, −7.951912875261990053582597622968, −7.44375960635166222944732063019, −6.17801494542483143369937597266, −4.85429871258052910201591878328, −3.68866242849198404595988976045, −3.10136977690793833510052680107, −1.12237424468986961176868979566,
0.54712928448343228198583327443, 2.69629893547749514731847645285, 4.20842937148244174874986818922, 4.71611260065913843435355536511, 6.28117842072093014980664319632, 7.23714947677144242372894337093, 7.59490730177684338060315931985, 8.567135235959343606996136074984, 9.650020448223171381491981301246, 10.16955595983152807213696210558