L(s) = 1 | − 1.73i·5-s + (−2 + 1.73i)7-s + 3.46i·11-s − 3.46i·13-s − 1.73i·17-s + 2·19-s + 2.00·25-s − 6·29-s − 2·31-s + (2.99 + 3.46i)35-s + 37-s − 5.19i·41-s − 1.73i·43-s − 3·47-s + (1.00 − 6.92i)49-s + ⋯ |
L(s) = 1 | − 0.774i·5-s + (−0.755 + 0.654i)7-s + 1.04i·11-s − 0.960i·13-s − 0.420i·17-s + 0.458·19-s + 0.400·25-s − 1.11·29-s − 0.359·31-s + (0.507 + 0.585i)35-s + 0.164·37-s − 0.811i·41-s − 0.264i·43-s − 0.437·47-s + (0.142 − 0.989i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6706074851\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6706074851\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2 - 1.73i)T \) |
good | 5 | \( 1 + 1.73iT - 5T^{2} \) |
| 11 | \( 1 - 3.46iT - 11T^{2} \) |
| 13 | \( 1 + 3.46iT - 13T^{2} \) |
| 17 | \( 1 + 1.73iT - 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - T + 37T^{2} \) |
| 41 | \( 1 + 5.19iT - 41T^{2} \) |
| 43 | \( 1 + 1.73iT - 43T^{2} \) |
| 47 | \( 1 + 3T + 47T^{2} \) |
| 53 | \( 1 + 12T + 53T^{2} \) |
| 59 | \( 1 - 3T + 59T^{2} \) |
| 61 | \( 1 - 10.3iT - 61T^{2} \) |
| 67 | \( 1 + 10.3iT - 67T^{2} \) |
| 71 | \( 1 + 13.8iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 1.73iT - 79T^{2} \) |
| 83 | \( 1 + 9T + 83T^{2} \) |
| 89 | \( 1 + 6.92iT - 89T^{2} \) |
| 97 | \( 1 + 3.46iT - 97T^{2} \) |
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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
−8.538225596250738746071656699791, −7.65317891872947567198260653330, −7.02887744805862734279190386725, −6.02108975144052243175831722757, −5.32943563341249127788960798635, −4.69808879210362399669480077006, −3.59034317131380380292723925032, −2.72681400540064032077794680292, −1.63474306267788138417370552630, −0.21353598480165498711523389896,
1.30537098157031757089803790129, 2.67087672183296198655144691848, 3.45789320107128695360130772864, 4.08498856869721507486668329926, 5.25598620863187273958643884337, 6.23953783063163088546270124137, 6.65617777905017750899702143223, 7.41603312725463411136300208463, 8.198738774309401000854670376699, 9.138248026449053302342377060069