L(s) = 1 | + 1.48i·2-s + 1.67i·3-s − 0.193·4-s − 2.48·6-s − i·7-s + 2.67i·8-s + 0.193·9-s + 11-s − 0.324i·12-s + 1.67i·13-s + 1.48·14-s − 4.35·16-s − 0.324i·17-s + 0.287i·18-s − 3.61·19-s + ⋯ |
L(s) = 1 | + 1.04i·2-s + 0.967i·3-s − 0.0969·4-s − 1.01·6-s − 0.377i·7-s + 0.945i·8-s + 0.0646·9-s + 0.301·11-s − 0.0937i·12-s + 0.464i·13-s + 0.395·14-s − 1.08·16-s − 0.0787i·17-s + 0.0677i·18-s − 0.828·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.640526355\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.640526355\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 - T \) |
good | 2 | \( 1 - 1.48iT - 2T^{2} \) |
| 3 | \( 1 - 1.67iT - 3T^{2} \) |
| 13 | \( 1 - 1.67iT - 13T^{2} \) |
| 17 | \( 1 + 0.324iT - 17T^{2} \) |
| 19 | \( 1 + 3.61T + 19T^{2} \) |
| 23 | \( 1 - 0.806iT - 23T^{2} \) |
| 29 | \( 1 + 7.92T + 29T^{2} \) |
| 31 | \( 1 - 1.13T + 31T^{2} \) |
| 37 | \( 1 - 7.76iT - 37T^{2} \) |
| 41 | \( 1 - 8.21T + 41T^{2} \) |
| 43 | \( 1 - 5.11iT - 43T^{2} \) |
| 47 | \( 1 - 2.24iT - 47T^{2} \) |
| 53 | \( 1 - 10.4iT - 53T^{2} \) |
| 59 | \( 1 + 4.48T + 59T^{2} \) |
| 61 | \( 1 + 12.1T + 61T^{2} \) |
| 67 | \( 1 - 4.15iT - 67T^{2} \) |
| 71 | \( 1 + 0.775T + 71T^{2} \) |
| 73 | \( 1 + 8.32iT - 73T^{2} \) |
| 79 | \( 1 - 2.15T + 79T^{2} \) |
| 83 | \( 1 + 12.8iT - 83T^{2} \) |
| 89 | \( 1 - 6.05T + 89T^{2} \) |
| 97 | \( 1 - 18.0iT - 97T^{2} \) |
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
−9.379904091639329159078508613668, −8.973135296436487664421370665393, −7.88269211894146952865463474042, −7.32437004282246106396041488524, −6.43932367580876565800168334871, −5.80626130741090332379692914395, −4.70226310696398100948086994489, −4.26984930185916525058381596874, −3.10039135952120189999380467156, −1.71459224453470299537584360572,
0.56736570261610887977654976841, 1.78912939549080234828832630251, 2.34602506402923540468496674495, 3.49881584024647219198461645668, 4.34571708813000142562300661354, 5.68672715444507203621409309248, 6.44510458233753340534571072786, 7.20406134188246387098064429314, 7.88052790240776668902941428923, 8.905951783443715531006662906720