| L(s) = 1 | − i·2-s − i·3-s − 4-s − 6-s − 5i·7-s + i·8-s + 2·9-s + 4·11-s + i·12-s − 3i·13-s − 5·14-s + 16-s + i·17-s − 2i·18-s + 2·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 0.408·6-s − 1.88i·7-s + 0.353i·8-s + 0.666·9-s + 1.20·11-s + 0.288i·12-s − 0.832i·13-s − 1.33·14-s + 0.250·16-s + 0.242i·17-s − 0.471i·18-s + 0.458·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.362006 - 1.53348i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.362006 - 1.53348i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 17 | \( 1 - iT \) |
| good | 3 | \( 1 + iT - 3T^{2} \) |
| 7 | \( 1 + 5iT - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 3iT - 13T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 8iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 5T + 31T^{2} \) |
| 37 | \( 1 + 12iT - 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 2iT - 47T^{2} \) |
| 53 | \( 1 - iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 - 5T + 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 - 17T + 79T^{2} \) |
| 83 | \( 1 - 16iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 16iT - 97T^{2} \) |
| 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
−9.935932653918886269536434220682, −9.272545328724082274114930237590, −7.901894787065130220673790992443, −7.35582596559058602402257256907, −6.59348847133954695692814000713, −5.24827592115097811521720728018, −3.94079454809184264529979870205, −3.60453979677846929911185308013, −1.68284938460488955088573347782, −0.859897077930414280487680904625,
1.84593902596164791082018060243, 3.33876123755406085082446524116, 4.52699866237421839301768852358, 5.20066124299851209927978087069, 6.34710373293401360465886975350, 6.82072579864366479006370627860, 8.217719875168666734990792292556, 9.002397941090564712783032827252, 9.356070953340746008689779836151, 10.26667828446045162003440933517
