| L(s) = 1 | + i·3-s + (1.48 + 1.67i)5-s − i·7-s − 9-s − 0.387·11-s + 2.96i·13-s + (−1.67 + 1.48i)15-s + 3.35i·17-s − 2.96·19-s + 21-s − 0.962i·23-s + (−0.612 + 4.96i)25-s − i·27-s − 1.22·29-s − 2.96·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (0.662 + 0.749i)5-s − 0.377i·7-s − 0.333·9-s − 0.116·11-s + 0.821i·13-s + (−0.432 + 0.382i)15-s + 0.812i·17-s − 0.679·19-s + 0.218·21-s − 0.200i·23-s + (−0.122 + 0.992i)25-s − 0.192i·27-s − 0.227·29-s − 0.532·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.662 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.458847712\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.458847712\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (-1.48 - 1.67i)T \) |
| 7 | \( 1 + iT \) |
| good | 11 | \( 1 + 0.387T + 11T^{2} \) |
| 13 | \( 1 - 2.96iT - 13T^{2} \) |
| 17 | \( 1 - 3.35iT - 17T^{2} \) |
| 19 | \( 1 + 2.96T + 19T^{2} \) |
| 23 | \( 1 + 0.962iT - 23T^{2} \) |
| 29 | \( 1 + 1.22T + 29T^{2} \) |
| 31 | \( 1 + 2.96T + 31T^{2} \) |
| 37 | \( 1 - 5.92iT - 37T^{2} \) |
| 41 | \( 1 - 1.03T + 41T^{2} \) |
| 43 | \( 1 - 10.7iT - 43T^{2} \) |
| 47 | \( 1 + 3.22iT - 47T^{2} \) |
| 53 | \( 1 - 5.66iT - 53T^{2} \) |
| 59 | \( 1 - 3.22T + 59T^{2} \) |
| 61 | \( 1 - 14.6T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 5.53T + 71T^{2} \) |
| 73 | \( 1 + 6.18iT - 73T^{2} \) |
| 79 | \( 1 + 13.9T + 79T^{2} \) |
| 83 | \( 1 - 3.22iT - 83T^{2} \) |
| 89 | \( 1 + 3.73T + 89T^{2} \) |
| 97 | \( 1 + 7.73iT - 97T^{2} \) |
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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
−9.794356389261152945571180931282, −8.968803425662126432300952902476, −8.152230698652909033678314632694, −7.10210523485312717754006984473, −6.41341815869977239177429424353, −5.67487280491658869331510574775, −4.57948093329126913116799050015, −3.79210960496064903669336808824, −2.75027625346384550231324048307, −1.66821510282946547914017341765,
0.53172716341158858796369340894, 1.86846097769975374467552395638, 2.73013217538304145810316858041, 4.04591397487981615724599830567, 5.31667146653945824828883130761, 5.59207124640889473382577472117, 6.65707714690848984445366711220, 7.49095275918662061189804218680, 8.391910593037542846878472286983, 8.942826112273657834054085522675
