L(s) = 1 | + 1.73·3-s + (0.5 − 0.866i)5-s + 1.99·9-s + i·11-s + (0.866 − 1.49i)15-s + i·23-s + (−0.499 − 0.866i)25-s + 1.73·27-s − 1.73·31-s + 1.73i·33-s + 37-s + (0.999 − 1.73i)45-s − 2i·47-s − 49-s − 2·53-s + ⋯ |
L(s) = 1 | + 1.73·3-s + (0.5 − 0.866i)5-s + 1.99·9-s + i·11-s + (0.866 − 1.49i)15-s + i·23-s + (−0.499 − 0.866i)25-s + 1.73·27-s − 1.73·31-s + 1.73i·33-s + 37-s + (0.999 − 1.73i)45-s − 2i·47-s − 49-s − 2·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.541380423\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.541380423\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 11 | \( 1 - iT \) |
good | 3 | \( 1 - 1.73T + T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - iT - T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + 1.73T + T^{2} \) |
| 37 | \( 1 - T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + 2iT - T^{2} \) |
| 53 | \( 1 + 2T + T^{2} \) |
| 59 | \( 1 - iT - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + 1.73T + T^{2} \) |
| 71 | \( 1 - 1.73T + T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T + T^{2} \) |
| 97 | \( 1 + 1.73iT - T^{2} \) |
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
−8.840149181825667865937764613234, −7.997759026862318412838619621605, −7.55406800618074320402592009285, −6.72606278455972047983013283388, −5.58547276837266319723095101618, −4.74076533436982496029330130184, −3.98487156476655116104784570174, −3.16411277429597276134549833025, −2.07644768176013178442443244347, −1.59015622500953324278208289862,
1.59586394564835957694595640638, 2.51984440025057587349426136577, 3.14911806263800228661888277752, 3.76040796602799370611806856003, 4.81838539709650204662610060526, 6.04070992569193969607619705728, 6.59829713238857716267700390804, 7.62917099194481546486592095932, 7.944267129135023836343935388069, 8.871137258366708326822622508311