| L(s) = 1 | + (2.5 − 1.32i)2-s + (4.5 − 6.61i)4-s + 18.5i·7-s + (2.5 − 22.4i)8-s − 27·9-s + 26.4i·11-s + (24.5 + 46.3i)14-s + (−23.5 − 59.5i)16-s + (−67.5 + 35.7i)18-s + (35 + 66.1i)22-s − 216. i·23-s + 125·25-s + (122. + 83.3i)28-s + 166·29-s + (−137.5 − 117. i)32-s + ⋯ |
| L(s) = 1 | + (0.883 − 0.467i)2-s + (0.562 − 0.826i)4-s + 0.999i·7-s + (0.110 − 0.993i)8-s − 9-s + 0.725i·11-s + (0.467 + 0.883i)14-s + (−0.367 − 0.930i)16-s + (−0.883 + 0.467i)18-s + (0.339 + 0.640i)22-s − 1.96i·23-s + 25-s + (0.826 + 0.562i)28-s + 1.06·29-s + (−0.759 − 0.650i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.826 + 0.562i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.826 + 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.67237 - 0.514951i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.67237 - 0.514951i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2.5 + 1.32i)T \) |
| 7 | \( 1 - 18.5iT \) |
| good | 3 | \( 1 + 27T^{2} \) |
| 5 | \( 1 - 125T^{2} \) |
| 11 | \( 1 - 26.4iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 2.19e3T^{2} \) |
| 17 | \( 1 - 4.91e3T^{2} \) |
| 19 | \( 1 + 6.85e3T^{2} \) |
| 23 | \( 1 + 216. iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 166T + 2.43e4T^{2} \) |
| 31 | \( 1 + 2.97e4T^{2} \) |
| 37 | \( 1 + 450T + 5.06e4T^{2} \) |
| 41 | \( 1 - 6.89e4T^{2} \) |
| 43 | \( 1 - 534. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 1.03e5T^{2} \) |
| 53 | \( 1 - 590T + 1.48e5T^{2} \) |
| 59 | \( 1 + 2.05e5T^{2} \) |
| 61 | \( 1 - 2.26e5T^{2} \) |
| 67 | \( 1 - 809. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 978. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 3.89e5T^{2} \) |
| 79 | \( 1 + 238. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 5.71e5T^{2} \) |
| 89 | \( 1 - 7.04e5T^{2} \) |
| 97 | \( 1 - 9.12e5T^{2} \) |
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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
−16.34874616429379835587257917946, −15.04334284307335384620963560684, −14.26097431025590630275121778250, −12.67101821545341736525998521168, −11.83946205848986778224486677400, −10.44240153339745126356822246640, −8.756510718150504070596558007508, −6.45097191580134934642806104348, −4.95504488697011621121989365465, −2.66561517619534552081322782857,
3.46714946368937619316805036946, 5.41970077640612376724332984629, 7.03334230120737163634274273655, 8.522927623651052168678769828541, 10.76047150926189384596856759000, 11.92833156374496356710178511261, 13.54993775569532649110878885228, 14.13770083321278279876189366394, 15.57021207718315343897723480586, 16.77281248002648551169840318575
