| L(s) = 1 | − i·3-s + 3.46·7-s + 2·9-s + 3i·11-s + 3.46i·13-s + 3·17-s + i·19-s − 3.46i·21-s − 5i·27-s + 10.3i·29-s − 6.92·31-s + 3·33-s + 10.3i·37-s + 3.46·39-s − 9·41-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 1.30·7-s + 0.666·9-s + 0.904i·11-s + 0.960i·13-s + 0.727·17-s + 0.229i·19-s − 0.755i·21-s − 0.962i·27-s + 1.92i·29-s − 1.24·31-s + 0.522·33-s + 1.70i·37-s + 0.554·39-s − 1.40·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.117499163\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.117499163\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + iT - 3T^{2} \) |
| 7 | \( 1 - 3.46T + 7T^{2} \) |
| 11 | \( 1 - 3iT - 11T^{2} \) |
| 13 | \( 1 - 3.46iT - 13T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 - iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 10.3iT - 29T^{2} \) |
| 31 | \( 1 + 6.92T + 31T^{2} \) |
| 37 | \( 1 - 10.3iT - 37T^{2} \) |
| 41 | \( 1 + 9T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 12iT - 59T^{2} \) |
| 61 | \( 1 + 3.46iT - 61T^{2} \) |
| 67 | \( 1 + 11iT - 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 - 7T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 + 15iT - 83T^{2} \) |
| 89 | \( 1 + 3T + 89T^{2} \) |
| 97 | \( 1 - 14T + 97T^{2} \) |
| 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
−9.407275345645034188468451584503, −8.538674684970453986735079609343, −7.69982657565425845104969500965, −7.17627607708531961793185974096, −6.41551232282068268546429082149, −5.08884856980941655817867113235, −4.64932629008631575130717058640, −3.49546605153785073194898558567, −1.87372056865281661912986566625, −1.48570495003332053803400892849,
0.922714657948824696762264671392, 2.25146763004207284532689741462, 3.58047722835624578966981436010, 4.28888829682471120431544748555, 5.35140195699131014234739318156, 5.76206524696414855628287636074, 7.22200466481087144536722102451, 7.80770547815864082042674784209, 8.573346421856997462989829503747, 9.355939795350531021465072082552
