L(s) = 1 | + 3-s + (1 − 2i)5-s + 2i·7-s + 9-s + 6i·11-s + 2·13-s + (1 − 2i)15-s − 6i·17-s − 4i·19-s + 2i·21-s + 8i·23-s + (−3 − 4i)25-s + 27-s + 8·31-s + 6i·33-s + ⋯ |
L(s) = 1 | + 0.577·3-s + (0.447 − 0.894i)5-s + 0.755i·7-s + 0.333·9-s + 1.80i·11-s + 0.554·13-s + (0.258 − 0.516i)15-s − 1.45i·17-s − 0.917i·19-s + 0.436i·21-s + 1.66i·23-s + (−0.600 − 0.800i)25-s + 0.192·27-s + 1.43·31-s + 1.04i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 - 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.723753614\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.723753614\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + (-1 + 2i)T \) |
good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 - 6iT - 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 - 8iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 4iT - 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 6iT - 59T^{2} \) |
| 61 | \( 1 - 6iT - 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 - 12iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 - 8iT - 97T^{2} \) |
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
−8.733820273216914718701308146871, −7.77353422828751937920084895569, −7.21551074569229676937747824172, −6.33824499941308818949989329903, −5.28696956116940240275729054352, −4.86428778680526759388612534048, −4.03516442239569406157107829835, −2.76891566852514825993641777777, −2.12576054785698034647156900768, −1.10440756254888688392528701383,
0.854119067424763064850588944514, 2.02482757013089088582550482867, 3.09016986958976323889115787108, 3.62159602338295760947142595201, 4.39189430287182709092506074615, 5.84875309917937703041865679390, 6.17459060460245651449405104106, 6.88999540728944346452539273545, 7.920632073914627816261596715370, 8.382924299285156204049586835220