| L(s) = 1 | + (0.866 − 1.5i)3-s + (1.73 − 2i)7-s + (−1.5 − 2.59i)9-s − 5.19i·11-s + 1.73·13-s − 3i·17-s + 3.46i·19-s + (−1.50 − 4.33i)21-s − 5.19·27-s + 5.19i·29-s − 10.3i·31-s + (−7.79 − 4.5i)33-s + 8i·37-s + (1.49 − 2.59i)39-s + 6·41-s + ⋯ |
| L(s) = 1 | + (0.499 − 0.866i)3-s + (0.654 − 0.755i)7-s + (−0.5 − 0.866i)9-s − 1.56i·11-s + 0.480·13-s − 0.727i·17-s + 0.794i·19-s + (−0.327 − 0.944i)21-s − 1.00·27-s + 0.964i·29-s − 1.86i·31-s + (−1.35 − 0.783i)33-s + 1.31i·37-s + (0.240 − 0.416i)39-s + 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.698 + 0.715i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.698 + 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.061122271\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.061122271\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.866 + 1.5i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.73 + 2i)T \) |
| good | 11 | \( 1 + 5.19iT - 11T^{2} \) |
| 13 | \( 1 - 1.73T + 13T^{2} \) |
| 17 | \( 1 + 3iT - 17T^{2} \) |
| 19 | \( 1 - 3.46iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 5.19iT - 29T^{2} \) |
| 31 | \( 1 + 10.3iT - 31T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 10iT - 43T^{2} \) |
| 47 | \( 1 + 3iT - 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 - 6.92iT - 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 + 10.3iT - 71T^{2} \) |
| 73 | \( 1 + 6.92T + 73T^{2} \) |
| 79 | \( 1 - 13T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 1.73T + 97T^{2} \) |
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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
−8.653160734958764936711260987949, −7.893290589610554699895742685948, −7.56775292261904460878497739582, −6.37594631072573060850254402659, −5.94399008594959295935062732965, −4.73327546028114120813746886896, −3.63237692079356897697981360001, −2.94567950912736010126619431626, −1.60236711526487201386561831856, −0.68871711143684570494167584604,
1.79002007350174635025350907170, 2.55040818087525135697579554133, 3.74016330093382265271055306653, 4.59170601540536594906036705976, 5.15915218431797337919316641110, 6.12368462959696060810076099141, 7.24072628393602974894512099908, 7.977040083004168235128825363467, 8.797809742005339142062412227423, 9.253879615449715385203872641382
