L(s) = 1 | + 1.37i·2-s − 3i·3-s + 6.10·4-s + 4.12·6-s + 7i·7-s + 19.4i·8-s − 9·9-s + 15.2·11-s − 18.3i·12-s − 76.7i·13-s − 9.63·14-s + 22.1·16-s − 96.7i·17-s − 12.3i·18-s + 14.1·19-s + ⋯ |
L(s) = 1 | + 0.486i·2-s − 0.577i·3-s + 0.763·4-s + 0.280·6-s + 0.377i·7-s + 0.857i·8-s − 0.333·9-s + 0.418·11-s − 0.440i·12-s − 1.63i·13-s − 0.183·14-s + 0.345·16-s − 1.37i·17-s − 0.162i·18-s + 0.171·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.461987953\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.461987953\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 3iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 7iT \) |
good | 2 | \( 1 - 1.37iT - 8T^{2} \) |
| 11 | \( 1 - 15.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 76.7iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 96.7iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 14.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 75.7iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 89.9T + 2.43e4T^{2} \) |
| 31 | \( 1 - 289.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 14.2iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 318.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 389. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 228. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 679. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 398.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 146.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 291. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 333.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 891. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 416.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 814. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 650.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.58e3iT - 9.12e5T^{2} \) |
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
−10.46283642104294008107252572739, −9.416486868471503021406790625721, −8.262804451114877317583255377168, −7.61687084098038135321477489561, −6.80437893050605341958058842295, −5.84168714831764970580184769516, −5.13560336206100503910500276209, −3.24226241701827172452644867538, −2.33362046304417805895279085841, −0.816391722085250691354384788376,
1.26540969360033535000070336614, 2.45602196428391404100093843614, 3.79240355175365033152429435516, 4.45140054116908207450207521880, 6.10663827929117703583051515354, 6.68719307144106017966583492903, 7.84224071745513322243202703381, 8.967126719455028923366501743529, 9.819112557675717858328101948638, 10.59558394431659182323138815576