| L(s) = 1 | + 11.0i·2-s − 52.4·3-s + 6.57·4-s + 215. i·5-s − 577. i·6-s − 343i·7-s + 1.48e3i·8-s + 561.·9-s − 2.37e3·10-s − 188. i·11-s − 344.·12-s + (−5.92e3 + 5.25e3i)13-s + 3.77e3·14-s − 1.12e4i·15-s − 1.54e4·16-s − 2.52e4·17-s + ⋯ |
| L(s) = 1 | + 0.973i·2-s − 1.12·3-s + 0.0513·4-s + 0.770i·5-s − 1.09i·6-s − 0.377i·7-s + 1.02i·8-s + 0.256·9-s − 0.750·10-s − 0.0427i·11-s − 0.0575·12-s + (−0.748 + 0.663i)13-s + 0.368·14-s − 0.864i·15-s − 0.946·16-s − 1.24·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.663 + 0.748i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.663 + 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.203248 - 0.0914418i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.203248 - 0.0914418i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + 343iT \) |
| 13 | \( 1 + (5.92e3 - 5.25e3i)T \) |
| good | 2 | \( 1 - 11.0iT - 128T^{2} \) |
| 3 | \( 1 + 52.4T + 2.18e3T^{2} \) |
| 5 | \( 1 - 215. iT - 7.81e4T^{2} \) |
| 11 | \( 1 + 188. iT - 1.94e7T^{2} \) |
| 17 | \( 1 + 2.52e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.34e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 + 7.18e3T + 3.40e9T^{2} \) |
| 29 | \( 1 + 4.52e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 5.93e4iT - 2.75e10T^{2} \) |
| 37 | \( 1 + 6.00e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 + 2.14e5iT - 1.94e11T^{2} \) |
| 43 | \( 1 - 8.28e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 9.33e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 - 9.37e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.50e6iT - 2.48e12T^{2} \) |
| 61 | \( 1 + 1.40e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.46e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 + 7.04e5iT - 9.09e12T^{2} \) |
| 73 | \( 1 + 4.16e6iT - 1.10e13T^{2} \) |
| 79 | \( 1 + 6.10e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 2.40e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 + 2.61e6iT - 4.42e13T^{2} \) |
| 97 | \( 1 + 5.95e5iT - 8.07e13T^{2} \) |
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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
−12.37940101366407302670522449059, −11.14435026803740459625325284644, −10.84515296965552976327592345847, −9.098017275226108743552538906540, −7.44149695538750121657447498462, −6.73532034346593982473744879505, −5.83719191725042948760471911717, −4.56514845858134782950397317646, −2.38963232125918218306108314472, −0.088420525515878673417161645156,
1.13042505955312360579231576596, 2.65990097345365976562086465320, 4.48465994731169709172183011642, 5.64553399973801731814592077762, 6.89961733444482260986499543245, 8.627556351857335458464903753071, 9.942794549021245843184588277358, 10.88061286349530503473383839681, 11.80977566478868295850903590984, 12.41192585320855240746758464458
