L(s) = 1 | − i·3-s + 0.0802·7-s − 9-s − 2.41i·11-s + 5.26i·13-s + 0.255·17-s + 6.95i·19-s − 0.0802i·21-s − 1.64·23-s + i·27-s − 4.51i·29-s − 8.29·31-s − 2.41·33-s + 2.67i·37-s + 5.26·39-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + 0.0303·7-s − 0.333·9-s − 0.728i·11-s + 1.46i·13-s + 0.0620·17-s + 1.59i·19-s − 0.0175i·21-s − 0.343·23-s + 0.192i·27-s − 0.838i·29-s − 1.48·31-s − 0.420·33-s + 0.439i·37-s + 0.843·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.136 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.136 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.040010338\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.040010338\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 0.0802T + 7T^{2} \) |
| 11 | \( 1 + 2.41iT - 11T^{2} \) |
| 13 | \( 1 - 5.26iT - 13T^{2} \) |
| 17 | \( 1 - 0.255T + 17T^{2} \) |
| 19 | \( 1 - 6.95iT - 19T^{2} \) |
| 23 | \( 1 + 1.64T + 23T^{2} \) |
| 29 | \( 1 + 4.51iT - 29T^{2} \) |
| 31 | \( 1 + 8.29T + 31T^{2} \) |
| 37 | \( 1 - 2.67iT - 37T^{2} \) |
| 41 | \( 1 + 8.11T + 41T^{2} \) |
| 43 | \( 1 - 4.08iT - 43T^{2} \) |
| 47 | \( 1 - 5.70T + 47T^{2} \) |
| 53 | \( 1 - 11.5iT - 53T^{2} \) |
| 59 | \( 1 - 12.6iT - 59T^{2} \) |
| 61 | \( 1 + 11.9iT - 61T^{2} \) |
| 67 | \( 1 - 7.27iT - 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 + 5.50T + 79T^{2} \) |
| 83 | \( 1 - 9.20iT - 83T^{2} \) |
| 89 | \( 1 - 11.9T + 89T^{2} \) |
| 97 | \( 1 + 8.50T + 97T^{2} \) |
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
−9.087923365550609470885841169996, −8.265668104624867231392309405531, −7.68753908312201137154048448885, −6.75373900320817096124749634846, −6.13006262225531311111015106265, −5.38257511459451706316743679424, −4.20315559566711379918423185383, −3.47217533483349097692792525556, −2.21031535350270520153198955508, −1.34622504134334371555255182409,
0.34916331711901678275310704190, 2.00796585341852114445123488342, 3.09101451677747958604180262011, 3.85626178617468616096600478313, 5.09753338359788707682430377785, 5.25175103040719851629138260654, 6.52564561791112578420239152610, 7.26721119885483287716504205875, 8.070294526222872973653747838949, 8.880598560122910963597979739048