| L(s) = 1 | + (−2.82 − i)3-s + 5.65·5-s − 6·7-s + (7.00 + 5.65i)9-s − 5.65·11-s + 10i·13-s + (−16.0 − 5.65i)15-s − 22.6i·17-s − 2i·19-s + (16.9 + 6i)21-s − 11.3i·23-s + 7.00·25-s + (−14.1 − 23.0i)27-s − 16.9·29-s + 22·31-s + ⋯ |
| L(s) = 1 | + (−0.942 − 0.333i)3-s + 1.13·5-s − 0.857·7-s + (0.777 + 0.628i)9-s − 0.514·11-s + 0.769i·13-s + (−1.06 − 0.377i)15-s − 1.33i·17-s − 0.105i·19-s + (0.808 + 0.285i)21-s − 0.491i·23-s + 0.280·25-s + (−0.523 − 0.851i)27-s − 0.585·29-s + 0.709·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.902 + 0.430i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.902 + 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4797506200\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4797506200\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.82 + i)T \) |
| good | 5 | \( 1 - 5.65T + 25T^{2} \) |
| 7 | \( 1 + 6T + 49T^{2} \) |
| 11 | \( 1 + 5.65T + 121T^{2} \) |
| 13 | \( 1 - 10iT - 169T^{2} \) |
| 17 | \( 1 + 22.6iT - 289T^{2} \) |
| 19 | \( 1 + 2iT - 361T^{2} \) |
| 23 | \( 1 + 11.3iT - 529T^{2} \) |
| 29 | \( 1 + 16.9T + 841T^{2} \) |
| 31 | \( 1 - 22T + 961T^{2} \) |
| 37 | \( 1 - 6iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 33.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 82iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 67.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 62.2T + 2.80e3T^{2} \) |
| 59 | \( 1 + 73.5T + 3.48e3T^{2} \) |
| 61 | \( 1 + 86iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 2iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 124. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 82T + 5.32e3T^{2} \) |
| 79 | \( 1 + 10T + 6.24e3T^{2} \) |
| 83 | \( 1 + 73.5T + 6.88e3T^{2} \) |
| 89 | \( 1 - 33.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 94T + 9.40e3T^{2} \) |
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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
−9.717909826432693845549925627988, −9.306816586208403597349814995799, −7.85457745139137474098217410054, −6.79738202810207090298406923215, −6.31030830261777643873984525689, −5.39816111247757074178685494784, −4.56344913750974262061837095086, −2.89422824708582426382274662847, −1.74485353870386351232963488559, −0.18203311097186505170605348271,
1.47641504114003126758450366172, 2.98000434247607352247063012204, 4.18903336733795748348759670342, 5.49903549427804157577999915932, 5.89296667811116547307356361554, 6.68433079768572424276658374055, 7.84628583713196070070171841483, 9.082953825467842733620406497803, 9.926546273132719889943719292978, 10.30659683525217098132264658476
