| L(s) = 1 | − 1.41i·2-s − 3.36i·3-s − 2.00·4-s − 4.76·6-s + 1.16·7-s + 2.82i·8-s − 2.32·9-s + 10.6i·11-s + 6.73i·12-s − 15.0i·13-s − 1.65i·14-s + 4.00·16-s + 20.0·17-s + 3.29i·18-s − 22.5i·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 1.12i·3-s − 0.500·4-s − 0.793·6-s + 0.167·7-s + 0.353i·8-s − 0.258·9-s + 0.964i·11-s + 0.560i·12-s − 1.16i·13-s − 0.118i·14-s + 0.250·16-s + 1.18·17-s + 0.183i·18-s − 1.18i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.775 - 0.631i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.775 - 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.209778107\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.209778107\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 5 | \( 1 \) |
| 23 | \( 1 + (9.45 + 20.9i)T \) |
| good | 3 | \( 1 + 3.36iT - 9T^{2} \) |
| 7 | \( 1 - 1.16T + 49T^{2} \) |
| 11 | \( 1 - 10.6iT - 121T^{2} \) |
| 13 | \( 1 + 15.0iT - 169T^{2} \) |
| 17 | \( 1 - 20.0T + 289T^{2} \) |
| 19 | \( 1 + 22.5iT - 361T^{2} \) |
| 29 | \( 1 + 32.5T + 841T^{2} \) |
| 31 | \( 1 + 27.0T + 961T^{2} \) |
| 37 | \( 1 + 53.0T + 1.36e3T^{2} \) |
| 41 | \( 1 - 9.43T + 1.68e3T^{2} \) |
| 43 | \( 1 + 36.4T + 1.84e3T^{2} \) |
| 47 | \( 1 - 49.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 104.T + 2.80e3T^{2} \) |
| 59 | \( 1 + 53.5T + 3.48e3T^{2} \) |
| 61 | \( 1 + 23.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 59.4T + 4.48e3T^{2} \) |
| 71 | \( 1 - 55.2T + 5.04e3T^{2} \) |
| 73 | \( 1 + 8.77iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 57.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 55.1T + 6.88e3T^{2} \) |
| 89 | \( 1 + 139. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 19.8T + 9.40e3T^{2} \) |
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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
−9.182085311032819654605752854379, −8.140697669308708169077126675299, −7.53344756295167855855290788635, −6.78868557485422221478223459201, −5.62981560581895117154748692194, −4.78420686212295326206383179680, −3.53853889080974981079905911812, −2.43184465904716726878073483961, −1.52447601296857199244105940253, −0.36974493976369487704400732826,
1.59885664693372394621864137205, 3.66091535309344876728075602198, 3.79692495378872572906536108366, 5.22982315607387608623809485930, 5.60795963467327682721068642478, 6.77262656289079412615841804969, 7.69145782610931697342400779224, 8.531339108822157560655477311679, 9.328880085906991151066109885568, 9.908676342882510210044953572110
