L(s) = 1 | − 3-s + 9-s + 1.73i·13-s − 19-s − 25-s − 27-s + 31-s + 37-s − 1.73i·39-s + 1.73i·43-s + 57-s + 1.73i·67-s + 1.73i·73-s + 75-s + 1.73i·79-s + ⋯ |
L(s) = 1 | − 3-s + 9-s + 1.73i·13-s − 19-s − 25-s − 27-s + 31-s + 37-s − 1.73i·39-s + 1.73i·43-s + 57-s + 1.73i·67-s + 1.73i·73-s + 75-s + 1.73i·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.188 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.188 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6974940110\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6974940110\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 - 1.73iT - T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T + T^{2} \) |
| 37 | \( 1 - T + T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - 1.73iT - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 1.73iT - T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 - 1.73iT - T^{2} \) |
| 79 | \( 1 - 1.73iT - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.561270677503805122355498127439, −8.580616281445769283078165628490, −7.71305952099040439551703764777, −6.74082661981307605698247453557, −6.36745349331958052283105433998, −5.48990342263638676624315600201, −4.38516185517913566818394619067, −4.11968271461581993850576373944, −2.46380859867583483318890949769, −1.37557283099724187685215143061,
0.56972120470717488003431296185, 2.04506623390400058431833018549, 3.31429906632349366715179955226, 4.32072814752031782033117745321, 5.14378935307829044279684937249, 5.92008102714748470895582333305, 6.44846584921669495121452682915, 7.54067486386295911316731889946, 8.032578093317050911805226551568, 9.049289727976366684166746688305