| L(s) = 1 | + (−3 + 4i)5-s + (−8 − 8i)7-s − 4·11-s + (−3 + 3i)13-s + (19 + 19i)17-s + 8i·19-s + (20 − 20i)23-s + (−7 − 24i)25-s − 38i·29-s + 44·31-s + (56 − 8i)35-s + (−3 − 3i)37-s + 70·41-s + (−36 + 36i)43-s + 79i·49-s + ⋯ |
| L(s) = 1 | + (−0.600 + 0.800i)5-s + (−1.14 − 1.14i)7-s − 0.363·11-s + (−0.230 + 0.230i)13-s + (1.11 + 1.11i)17-s + 0.421i·19-s + (0.869 − 0.869i)23-s + (−0.280 − 0.959i)25-s − 1.31i·29-s + 1.41·31-s + (1.59 − 0.228i)35-s + (−0.0810 − 0.0810i)37-s + 1.70·41-s + (−0.837 + 0.837i)43-s + 1.61i·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0898i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.995 + 0.0898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.254208471\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.254208471\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (3 - 4i)T \) |
| good | 7 | \( 1 + (8 + 8i)T + 49iT^{2} \) |
| 11 | \( 1 + 4T + 121T^{2} \) |
| 13 | \( 1 + (3 - 3i)T - 169iT^{2} \) |
| 17 | \( 1 + (-19 - 19i)T + 289iT^{2} \) |
| 19 | \( 1 - 8iT - 361T^{2} \) |
| 23 | \( 1 + (-20 + 20i)T - 529iT^{2} \) |
| 29 | \( 1 + 38iT - 841T^{2} \) |
| 31 | \( 1 - 44T + 961T^{2} \) |
| 37 | \( 1 + (3 + 3i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 - 70T + 1.68e3T^{2} \) |
| 43 | \( 1 + (36 - 36i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + 2.20e3iT^{2} \) |
| 53 | \( 1 + (-17 + 17i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 - 92iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 72T + 3.72e3T^{2} \) |
| 67 | \( 1 + (44 + 44i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 - 88T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-55 + 55i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 + 12iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-24 + 24i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 - 26iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (57 + 57i)T + 9.40e3iT^{2} \) |
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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
−10.25855166908221144521464115286, −9.658674429345818461754997684760, −8.217909497037077241467033913256, −7.57773312420896442957282389244, −6.67894433801318225971474379019, −6.03878377034099197973409687719, −4.41513989746783041795131358779, −3.62773265149013519630787397340, −2.70715574144558870786913655495, −0.69477196392429294835566447200,
0.78390197828119058250098183436, 2.68242743396678021555928233445, 3.48791375170214335022265567108, 5.02868733567091205457230949929, 5.49195499037577199538527893043, 6.75792180828812070321648921523, 7.66054454318612364454405280357, 8.636451523012043204044071442020, 9.341878529635613317592889806872, 9.945337734104073528965330337572
