L(s) = 1 | + 2i·3-s + 2i·7-s − 9-s + 2i·13-s + 6i·17-s − 4·19-s − 4·21-s − 6i·23-s + 4i·27-s − 6·29-s + 4·31-s − 2i·37-s − 4·39-s + 6·41-s + 10i·43-s + ⋯ |
L(s) = 1 | + 1.15i·3-s + 0.755i·7-s − 0.333·9-s + 0.554i·13-s + 1.45i·17-s − 0.917·19-s − 0.872·21-s − 1.25i·23-s + 0.769i·27-s − 1.11·29-s + 0.718·31-s − 0.328i·37-s − 0.640·39-s + 0.937·41-s + 1.52i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.664050 + 1.07445i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.664050 + 1.07445i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - 2iT - 3T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 10iT - 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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
−11.32734545736365554889082141083, −10.59635395971337236071032249350, −9.799071690705238471210680610521, −8.909843331188800465642844077057, −8.214104276581169430209889968862, −6.67727215748026356498706456766, −5.72290744922611099450700787730, −4.55694476169128296686606688384, −3.77527774980063094396502606883, −2.20514001477767297587504975781,
0.857840239457185337066244292070, 2.37040898097720189295123822923, 3.87648931552195386388538551592, 5.26127125975652623331887073362, 6.45698248089948531844922588478, 7.34582026778733716604996637182, 7.82734651441665781875905466194, 9.109737120649100099728257769833, 10.11479236960394371978330958967, 11.11145598250743589463813941637