| L(s) = 1 | − i·2-s + i·3-s − 4-s + 6-s − i·7-s + i·8-s + 2·9-s + 3·11-s − i·12-s + 2i·13-s − 14-s + 16-s − 3i·17-s − 2i·18-s + 7·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.408·6-s − 0.377i·7-s + 0.353i·8-s + 0.666·9-s + 0.904·11-s − 0.288i·12-s + 0.554i·13-s − 0.267·14-s + 0.250·16-s − 0.727i·17-s − 0.471i·18-s + 1.60·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.36870 - 0.323107i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.36870 - 0.323107i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 + iT \) |
| good | 3 | \( 1 - iT - 3T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 3iT - 17T^{2} \) |
| 19 | \( 1 - 7T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 + 9T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 + 12iT - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 7iT - 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - 5iT - 73T^{2} \) |
| 79 | \( 1 + 14T + 79T^{2} \) |
| 83 | \( 1 + 9iT - 83T^{2} \) |
| 89 | \( 1 - 15T + 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.43542080794687841361161320888, −10.47171892529940308758171697404, −9.579009428886797063318045648029, −9.137854964217652677403368906993, −7.66578944834140103048419730306, −6.67972868613410665024264603752, −5.12974229193952936245223353764, −4.22140969165292187868312288693, −3.22476268795815471130630124211, −1.38643737698553398363216844504,
1.39931523367926616697580766515, 3.38090030408863777519563102748, 4.75532718895897824441470639549, 5.92173721221202844095837243996, 6.81224159074683869205437025071, 7.64782974353942160188456327130, 8.585608861462356437225398783181, 9.568685522922711961902715967641, 10.47158512922332734982283242374, 11.92881269295989137345621351552
