| L(s) = 1 | + 46.7i·3-s + 283. i·5-s + (−1.23e3 + 2.06e3i)7-s − 2.18e3·9-s + 9.37e3·11-s − 3.95e4i·13-s − 1.32e4·15-s + 1.00e5i·17-s − 2.83e4i·19-s + (−9.64e4 − 5.75e4i)21-s + 2.56e5·23-s + 3.10e5·25-s − 1.02e5i·27-s − 2.03e5·29-s + 1.53e6i·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.453i·5-s + (−0.512 + 0.858i)7-s − 0.333·9-s + 0.640·11-s − 1.38i·13-s − 0.261·15-s + 1.20i·17-s − 0.217i·19-s + (−0.495 − 0.295i)21-s + 0.915·23-s + 0.794·25-s − 0.192i·27-s − 0.288·29-s + 1.65i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.858 - 0.512i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.858 - 0.512i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.769899470\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.769899470\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 46.7iT \) |
| 7 | \( 1 + (1.23e3 - 2.06e3i)T \) |
| good | 5 | \( 1 - 283. iT - 3.90e5T^{2} \) |
| 11 | \( 1 - 9.37e3T + 2.14e8T^{2} \) |
| 13 | \( 1 + 3.95e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 - 1.00e5iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 2.83e4iT - 1.69e10T^{2} \) |
| 23 | \( 1 - 2.56e5T + 7.83e10T^{2} \) |
| 29 | \( 1 + 2.03e5T + 5.00e11T^{2} \) |
| 31 | \( 1 - 1.53e6iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 2.65e6T + 3.51e12T^{2} \) |
| 41 | \( 1 - 2.60e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 6.48e6T + 1.16e13T^{2} \) |
| 47 | \( 1 - 6.95e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 1.48e7T + 6.22e13T^{2} \) |
| 59 | \( 1 + 1.78e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 1.29e7iT - 1.91e14T^{2} \) |
| 67 | \( 1 + 3.01e7T + 4.06e14T^{2} \) |
| 71 | \( 1 - 3.09e7T + 6.45e14T^{2} \) |
| 73 | \( 1 - 2.54e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 - 3.23e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + 4.10e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 9.26e6iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 7.69e7iT - 7.83e15T^{2} \) |
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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
−10.63514526468892851574009456176, −9.595937198941222187493135744546, −8.849765934071770321393116000371, −7.87001610345971489531807876950, −6.52954418865172248402172903492, −5.80574983186268909640092471549, −4.70100780032008203839202275369, −3.35323321869528567991524486025, −2.74298838240296459075454294331, −1.11766195295980677757383921574,
0.41093823011887908215797969844, 1.20493834616435281839000097049, 2.48650439678582093274786676556, 3.84664545840837735804011630737, 4.74532245276948909191902579520, 6.14916144529238767393252225537, 6.97525813783286282082733870050, 7.68013060970487483894102082867, 9.137611578725343601838986796289, 9.433475457539635572564923892989
