L(s) = 1 | + 1.41i·3-s + 2.64·7-s + 0.999·9-s + 3.74i·11-s − 4.24i·13-s + 5.29·19-s + 3.74i·21-s + (3 − 3.74i)23-s − 5·25-s + 5.65i·27-s + 6·29-s + 8.48i·31-s − 5.29·33-s − 11.2i·37-s + 6·39-s + ⋯ |
L(s) = 1 | + 0.816i·3-s + 0.999·7-s + 0.333·9-s + 1.12i·11-s − 1.17i·13-s + 1.21·19-s + 0.816i·21-s + (0.625 − 0.780i)23-s − 25-s + 1.08i·27-s + 1.11·29-s + 1.52i·31-s − 0.921·33-s − 1.84i·37-s + 0.960·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.625 - 0.780i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.625 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.311116202\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.311116202\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - 2.64T \) |
| 23 | \( 1 + (-3 + 3.74i)T \) |
good | 3 | \( 1 - 1.41iT - 3T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 - 3.74iT - 11T^{2} \) |
| 13 | \( 1 + 4.24iT - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 5.29T + 19T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 8.48iT - 31T^{2} \) |
| 37 | \( 1 + 11.2iT - 37T^{2} \) |
| 41 | \( 1 - 5.65iT - 41T^{2} \) |
| 43 | \( 1 + 11.2iT - 43T^{2} \) |
| 47 | \( 1 - 2.82iT - 47T^{2} \) |
| 53 | \( 1 + 3.74iT - 53T^{2} \) |
| 59 | \( 1 + 9.89iT - 59T^{2} \) |
| 61 | \( 1 - 10.5T + 61T^{2} \) |
| 67 | \( 1 - 11.2iT - 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 - 8.48iT - 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 15.8T + 83T^{2} \) |
| 89 | \( 1 - 15.8T + 89T^{2} \) |
| 97 | \( 1 + 5.29T + 97T^{2} \) |
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
−9.023417951417864507615499745295, −8.284775894771633193920278377110, −7.45043269398359145531812674296, −6.93127238582756090319041990570, −5.50797313920747180941834411383, −5.05916529625712966512513547019, −4.33434567855740257558573644448, −3.45080002586962300937467625592, −2.30520733186659616881336764595, −1.10773179750183835997731644017,
0.996749849920824345862613272028, 1.73186654180911346047782604561, 2.87721432859700053196714074370, 4.03176660498491586558298498061, 4.83786593985071988520371110232, 5.80832192920134348607548072825, 6.50760327051521737751408480948, 7.41038935625392119995422296561, 7.87355378228929608250183733803, 8.626501790763395071084342045740