| L(s) = 1 | + (1.5 + 0.866i)3-s + 3.46i·5-s + (1.5 + 2.59i)9-s + 6·11-s + 13-s + (−2.99 + 5.19i)15-s − 3.46i·19-s − 23-s − 6.99·25-s + 5.19i·27-s − 8.66i·29-s + 5.19i·31-s + (9 + 5.19i)33-s − 10·37-s + (1.5 + 0.866i)39-s + ⋯ |
| L(s) = 1 | + (0.866 + 0.499i)3-s + 1.54i·5-s + (0.5 + 0.866i)9-s + 1.80·11-s + 0.277·13-s + (−0.774 + 1.34i)15-s − 0.794i·19-s − 0.208·23-s − 1.39·25-s + 0.999i·27-s − 1.60i·29-s + 0.933i·31-s + (1.56 + 0.904i)33-s − 1.64·37-s + (0.240 + 0.138i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.408974343\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.408974343\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.5 - 0.866i)T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 3.46iT - 5T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 6T + 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 3.46iT - 19T^{2} \) |
| 29 | \( 1 + 8.66iT - 29T^{2} \) |
| 31 | \( 1 - 5.19iT - 31T^{2} \) |
| 37 | \( 1 + 10T + 37T^{2} \) |
| 41 | \( 1 + 1.73iT - 41T^{2} \) |
| 43 | \( 1 - 6.92iT - 43T^{2} \) |
| 47 | \( 1 + 3T + 47T^{2} \) |
| 53 | \( 1 - 3.46iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 10.3iT - 67T^{2} \) |
| 71 | \( 1 - 15T + 71T^{2} \) |
| 73 | \( 1 + 11T + 73T^{2} \) |
| 79 | \( 1 + 3.46iT - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 13.8iT - 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−10.00492361915376468498839207318, −9.278351432480062948320972647455, −8.558510393918187539500697938154, −7.47303197697428084078917248361, −6.80701813540956025674305817243, −6.07829173868640636342764531872, −4.55033574332418493836495818921, −3.67408183018371487254636593499, −2.98582292278994128654270035810, −1.82996832114887022890382770621,
1.09752692984182557764900905155, 1.81412437721667386287496972268, 3.57054499890382741177332760189, 4.15291153691203318063209062094, 5.33129306933187001219292379385, 6.37533478700052996904955359690, 7.22910193236004253749420200227, 8.272308660510173025233607425095, 8.857365358301353970998507064639, 9.240181343680440530196874336760
