L(s) = 1 | + (−0.0384 − 1.73i)3-s + (−2.22 − 0.176i)5-s + (3.29 − 3.29i)7-s + (−2.99 + 0.132i)9-s + (−2.51 + 0.817i)11-s + (−2.61 + 1.33i)13-s + (−0.219 + 3.86i)15-s + (−1.07 − 6.77i)17-s + (0.761 − 1.04i)19-s + (−5.83 − 5.58i)21-s + (0.460 + 0.234i)23-s + (4.93 + 0.786i)25-s + (0.345 + 5.18i)27-s + (3.38 − 2.46i)29-s + (−6.95 − 5.05i)31-s + ⋯ |
L(s) = 1 | + (−0.0221 − 0.999i)3-s + (−0.996 − 0.0788i)5-s + (1.24 − 1.24i)7-s + (−0.999 + 0.0443i)9-s + (−0.758 + 0.246i)11-s + (−0.725 + 0.369i)13-s + (−0.0567 + 0.998i)15-s + (−0.260 − 1.64i)17-s + (0.174 − 0.240i)19-s + (−1.27 − 1.21i)21-s + (0.0959 + 0.0488i)23-s + (0.987 + 0.157i)25-s + (0.0664 + 0.997i)27-s + (0.629 − 0.457i)29-s + (−1.24 − 0.907i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.645 + 0.764i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.645 + 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.405040 - 0.872178i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.405040 - 0.872178i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.0384 + 1.73i)T \) |
| 5 | \( 1 + (2.22 + 0.176i)T \) |
good | 7 | \( 1 + (-3.29 + 3.29i)T - 7iT^{2} \) |
| 11 | \( 1 + (2.51 - 0.817i)T + (8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (2.61 - 1.33i)T + (7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (1.07 + 6.77i)T + (-16.1 + 5.25i)T^{2} \) |
| 19 | \( 1 + (-0.761 + 1.04i)T + (-5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-0.460 - 0.234i)T + (13.5 + 18.6i)T^{2} \) |
| 29 | \( 1 + (-3.38 + 2.46i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (6.95 + 5.05i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-0.868 - 1.70i)T + (-21.7 + 29.9i)T^{2} \) |
| 41 | \( 1 + (-6.79 - 2.20i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (-6.48 - 6.48i)T + 43iT^{2} \) |
| 47 | \( 1 + (-9.43 - 1.49i)T + (44.6 + 14.5i)T^{2} \) |
| 53 | \( 1 + (-0.368 + 2.32i)T + (-50.4 - 16.3i)T^{2} \) |
| 59 | \( 1 + (-0.216 + 0.664i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (1.56 + 4.80i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (-11.0 + 1.74i)T + (63.7 - 20.7i)T^{2} \) |
| 71 | \( 1 + (-3.02 - 4.16i)T + (-21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (1.39 - 2.74i)T + (-42.9 - 59.0i)T^{2} \) |
| 79 | \( 1 + (-1.41 - 1.94i)T + (-24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-0.525 + 0.0832i)T + (78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (-1.78 - 5.48i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-1.58 + 10.0i)T + (-92.2 - 29.9i)T^{2} \) |
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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
−11.34350660150566014821651453430, −10.95868669212755703774849207005, −9.420571821122323936728132241306, −8.062320333266126681658364918659, −7.53448512799744002097765182621, −7.00478724414987546697823205388, −5.17140872460950028229249825521, −4.29637428865373382536913907773, −2.53661717787635095692372329926, −0.71888028650778590459696379365,
2.51323850794350614679763154832, 3.90704177557408387370582835043, 5.02024391264836837319339459132, 5.74164410693706159000757157069, 7.59911082770871085892357661849, 8.422955730534324164497766128874, 9.007211277521307156465327381070, 10.62677014024148824695397613784, 10.88161028291015432350917774068, 12.04703347259618735011333720825