| L(s) = 1 | − i·3-s + 3.54·5-s − 5.04i·7-s − 9-s + 1.25i·11-s − 4.25i·13-s − 3.54i·15-s − 0.236i·17-s − 0.840i·19-s − 5.04·21-s − 7.74·23-s + 7.58·25-s + i·27-s + 10.3i·29-s − 4.69·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 1.58·5-s − 1.90i·7-s − 0.333·9-s + 0.378i·11-s − 1.18i·13-s − 0.916i·15-s − 0.0574i·17-s − 0.192i·19-s − 1.10·21-s − 1.61·23-s + 1.51·25-s + 0.192i·27-s + 1.92i·29-s − 0.843·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.147 + 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.147 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.27974 - 1.48484i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.27974 - 1.48484i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 41 | \( 1 + (-6.33 - 0.944i)T \) |
| good | 5 | \( 1 - 3.54T + 5T^{2} \) |
| 7 | \( 1 + 5.04iT - 7T^{2} \) |
| 11 | \( 1 - 1.25iT - 11T^{2} \) |
| 13 | \( 1 + 4.25iT - 13T^{2} \) |
| 17 | \( 1 + 0.236iT - 17T^{2} \) |
| 19 | \( 1 + 0.840iT - 19T^{2} \) |
| 23 | \( 1 + 7.74T + 23T^{2} \) |
| 29 | \( 1 - 10.3iT - 29T^{2} \) |
| 31 | \( 1 + 4.69T + 31T^{2} \) |
| 37 | \( 1 - 5.58T + 37T^{2} \) |
| 43 | \( 1 - 7.82T + 43T^{2} \) |
| 47 | \( 1 - 2.07iT - 47T^{2} \) |
| 53 | \( 1 - 1.29iT - 53T^{2} \) |
| 59 | \( 1 - 7.40T + 59T^{2} \) |
| 61 | \( 1 + 10.7T + 61T^{2} \) |
| 67 | \( 1 - 5.18iT - 67T^{2} \) |
| 71 | \( 1 + 6.74iT - 71T^{2} \) |
| 73 | \( 1 - 8.41T + 73T^{2} \) |
| 79 | \( 1 + 11.1iT - 79T^{2} \) |
| 83 | \( 1 - 10.0T + 83T^{2} \) |
| 89 | \( 1 + 3.21iT - 89T^{2} \) |
| 97 | \( 1 + 3.94iT - 97T^{2} \) |
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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
−9.936765120326557250012630849128, −9.090422032917134740449255216381, −7.81331993502962527409913221387, −7.29676061896446347336742173052, −6.34819965171001229047370220152, −5.62363041723832725174608589588, −4.52515202642055469924122339678, −3.27754685924198528306665982792, −1.97556763099070301924959740013, −0.910136864532009973443132854673,
2.05693856051699383596557936336, 2.46859545471333518426984876705, 4.06257477300678988992102878249, 5.25901459334615437110119600898, 6.03141239463936448039198657092, 6.21994547086711349152047873889, 7.973914429609208561282653473340, 8.909123698889684151822993835544, 9.457932782784956245253204263735, 9.847037379035664888108126451998
