L(s) = 1 | + 4.39i·5-s − 3.57·7-s − 14.6i·11-s + 3.91·13-s − 6.88i·17-s − 28.2·19-s + 1.29i·23-s + 5.68·25-s + 37.9i·29-s + 54.4·31-s − 15.7i·35-s − 37.0·37-s − 41.1i·41-s − 3.54·43-s + 42.8i·47-s + ⋯ |
L(s) = 1 | + 0.879i·5-s − 0.510·7-s − 1.33i·11-s + 0.301·13-s − 0.405i·17-s − 1.48·19-s + 0.0563i·23-s + 0.227·25-s + 1.31i·29-s + 1.75·31-s − 0.448i·35-s − 1.00·37-s − 1.00i·41-s − 0.0824·43-s + 0.910i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.309881696\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.309881696\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 4.39iT - 25T^{2} \) |
| 7 | \( 1 + 3.57T + 49T^{2} \) |
| 11 | \( 1 + 14.6iT - 121T^{2} \) |
| 13 | \( 1 - 3.91T + 169T^{2} \) |
| 17 | \( 1 + 6.88iT - 289T^{2} \) |
| 19 | \( 1 + 28.2T + 361T^{2} \) |
| 23 | \( 1 - 1.29iT - 529T^{2} \) |
| 29 | \( 1 - 37.9iT - 841T^{2} \) |
| 31 | \( 1 - 54.4T + 961T^{2} \) |
| 37 | \( 1 + 37.0T + 1.36e3T^{2} \) |
| 41 | \( 1 + 41.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 3.54T + 1.84e3T^{2} \) |
| 47 | \( 1 - 42.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 47.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 62.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 11.1T + 3.72e3T^{2} \) |
| 67 | \( 1 - 111.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 105. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 97.0T + 5.32e3T^{2} \) |
| 79 | \( 1 - 110.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 131. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 101. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 165.T + 9.40e3T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.576429518294882976678052306670, −8.259061511092070362513389499735, −6.96664860929045348826495040536, −6.61689676321906863983824433446, −5.89213202043205986117541949379, −4.95229498717642561203362603438, −3.80028871606183258470773285760, −3.15806405298262145028085968484, −2.36741822101282058328934994007, −0.896724616056180350296276347897,
0.35949084851012476333109155005, 1.62336711267269427287349062401, 2.50889121071162008622447749331, 3.78732600265374732547320683045, 4.53777148122197099899538783912, 5.10407336969019829005220713857, 6.40131893219104419465002243937, 6.58319557170166453664355806482, 7.87197229490183103459946311019, 8.336800839268723955646749129548