L(s) = 1 | + i·2-s + 4-s − i·5-s + 2·7-s + 3i·8-s − 3·9-s + 10-s + 2·11-s + 2i·13-s + 2i·14-s − 16-s − 3i·17-s − 3i·18-s + 6i·19-s − i·20-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 0.5·4-s − 0.447i·5-s + 0.755·7-s + 1.06i·8-s − 9-s + 0.316·10-s + 0.603·11-s + 0.554i·13-s + 0.534i·14-s − 0.250·16-s − 0.727i·17-s − 0.707i·18-s + 1.37i·19-s − 0.223i·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1369 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.164 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1369 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.164 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.084567210\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.084567210\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
good | 2 | \( 1 - iT - 2T^{2} \) |
| 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 + iT - 5T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 3iT - 17T^{2} \) |
| 19 | \( 1 - 6iT - 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 9iT - 29T^{2} \) |
| 31 | \( 1 + 10iT - 31T^{2} \) |
| 41 | \( 1 - 9T + 41T^{2} \) |
| 43 | \( 1 + 2iT - 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 - 4iT - 59T^{2} \) |
| 61 | \( 1 - iT - 61T^{2} \) |
| 67 | \( 1 - 10T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 + 10iT - 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 - 7iT - 89T^{2} \) |
| 97 | \( 1 + 7iT - 97T^{2} \) |
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
−9.469405250858333139856631744935, −8.817704877407745737109133813974, −8.037094102097097496330275314050, −7.41927212550470090055314017544, −6.46131305779098999461042783181, −5.64758926230507540537518691371, −5.04007458322309951563925848145, −3.83322059156717427375609800087, −2.54032857992879832688774485749, −1.40913665351903306798354418610,
0.918897384453553686228378030356, 2.32305371418684077330378160563, 2.96286592349509870531365920060, 4.05792798512623883656923640282, 5.15338786753145306060573166697, 6.26613157145651842087006719510, 6.82163947242747008542036682539, 7.895851122014243198290871213314, 8.627704879605355223176150547974, 9.518574035798762731419040068585