| L(s) = 1 | + 8i·2-s + 49.3i·3-s − 64·4-s − 394.·6-s + 343i·7-s − 512i·8-s − 243.·9-s − 3.77e3·11-s − 3.15e3i·12-s − 6.86e3i·13-s − 2.74e3·14-s + 4.09e3·16-s + 1.52e4i·17-s − 1.94e3i·18-s − 2.10e4·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 1.05i·3-s − 0.5·4-s − 0.745·6-s + 0.377i·7-s − 0.353i·8-s − 0.111·9-s − 0.856·11-s − 0.527i·12-s − 0.866i·13-s − 0.267·14-s + 0.250·16-s + 0.753i·17-s − 0.0787i·18-s − 0.702·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.567500054\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.567500054\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 8iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 343iT \) |
| good | 3 | \( 1 - 49.3iT - 2.18e3T^{2} \) |
| 11 | \( 1 + 3.77e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 6.86e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 - 1.52e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 + 2.10e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 9.24e4iT - 3.40e9T^{2} \) |
| 29 | \( 1 + 7.86e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.52e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.45e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 - 3.84e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 2.64e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 + 2.25e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 + 2.63e5iT - 1.17e12T^{2} \) |
| 59 | \( 1 + 9.43e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.40e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 4.18e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 - 5.10e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.16e6iT - 1.10e13T^{2} \) |
| 79 | \( 1 - 5.00e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 5.49e5iT - 2.71e13T^{2} \) |
| 89 | \( 1 - 3.34e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.54e6iT - 8.07e13T^{2} \) |
| 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
−10.42010739906097248348726043709, −9.513059004794196542177872892550, −8.534589046782032329551650776592, −7.81405066536903758055418708164, −6.48850210883995776802624341478, −5.51079409022242760297772670868, −4.67191753970485882366629250370, −3.71880819934961154682809656266, −2.39496923066067519558058989281, −0.46487262549327110866363308595,
0.75331753309352303785891048237, 1.72510030869690200513554946762, 2.67197266616893315015538703294, 4.01341486589629185275671842654, 5.13901202851118832812273705421, 6.44601968186503434112930024299, 7.37848734233560318917213515344, 8.129567912294613114275924385527, 9.341257085204859170348690606474, 10.17223243065254885148484548966
