| L(s) = 1 | + (1.48 + 1.67i)5-s − i·7-s + 2·11-s + 1.35i·13-s − 3.35i·17-s + 5.35·19-s + 4.96i·23-s + (−0.612 + 4.96i)25-s + 7.92·29-s − 4.57·31-s + (1.67 − 1.48i)35-s + 0.775i·37-s − 3.73·41-s + 12.6i·43-s − 9.92i·47-s + ⋯ |
| L(s) = 1 | + (0.662 + 0.749i)5-s − 0.377i·7-s + 0.603·11-s + 0.374i·13-s − 0.812i·17-s + 1.22·19-s + 1.03i·23-s + (−0.122 + 0.992i)25-s + 1.47·29-s − 0.821·31-s + (0.283 − 0.250i)35-s + 0.127i·37-s − 0.583·41-s + 1.92i·43-s − 1.44i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 - 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.437776860\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.437776860\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.48 - 1.67i)T \) |
| 7 | \( 1 + iT \) |
| good | 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 1.35iT - 13T^{2} \) |
| 17 | \( 1 + 3.35iT - 17T^{2} \) |
| 19 | \( 1 - 5.35T + 19T^{2} \) |
| 23 | \( 1 - 4.96iT - 23T^{2} \) |
| 29 | \( 1 - 7.92T + 29T^{2} \) |
| 31 | \( 1 + 4.57T + 31T^{2} \) |
| 37 | \( 1 - 0.775iT - 37T^{2} \) |
| 41 | \( 1 + 3.73T + 41T^{2} \) |
| 43 | \( 1 - 12.6iT - 43T^{2} \) |
| 47 | \( 1 + 9.92iT - 47T^{2} \) |
| 53 | \( 1 + 8.57iT - 53T^{2} \) |
| 59 | \( 1 - 8.62T + 59T^{2} \) |
| 61 | \( 1 + 8.70T + 61T^{2} \) |
| 67 | \( 1 - 9.92iT - 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 + 9.35iT - 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 - 3.22iT - 83T^{2} \) |
| 89 | \( 1 - 1.03T + 89T^{2} \) |
| 97 | \( 1 - 18.4iT - 97T^{2} \) |
| 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
−8.278662738586615926870896039657, −7.37896160497159028081820603201, −6.93100039586462271876081066034, −6.27365428169055680019302781998, −5.41219742335914311945055597597, −4.74597978841395650977079016012, −3.60571634337043927841808893470, −3.07158807484538320285607929457, −1.99525130872959086556673711418, −1.03890333790404018913158555316,
0.77812707746309052233231597719, 1.71121007533927141228239653474, 2.66795467240823144768163905718, 3.65186820106578845331241954470, 4.58292455433892381873948559355, 5.25364107252715784446907759435, 5.97577478866918298403727705753, 6.54199943623135376780016838461, 7.50009413081767379509368999382, 8.312584008016920911644393217453
