L(s) = 1 | − 5-s − 2.09i·7-s − 3.07·11-s + 3.07·13-s + 2.79·17-s − 7.09i·19-s + (−2.07 + 4.32i)23-s + 25-s − 7.02i·29-s + 2.54·31-s + 2.09i·35-s − 11.8i·37-s − 1.34i·41-s + 7.71i·43-s − 2.64i·47-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.790i·7-s − 0.927·11-s + 0.852·13-s + 0.678·17-s − 1.62i·19-s + (−0.433 + 0.901i)23-s + 0.200·25-s − 1.30i·29-s + 0.457·31-s + 0.353i·35-s − 1.94i·37-s − 0.210i·41-s + 1.17i·43-s − 0.385i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.874 + 0.485i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.874 + 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9616171633\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9616171633\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + (2.07 - 4.32i)T \) |
good | 7 | \( 1 + 2.09iT - 7T^{2} \) |
| 11 | \( 1 + 3.07T + 11T^{2} \) |
| 13 | \( 1 - 3.07T + 13T^{2} \) |
| 17 | \( 1 - 2.79T + 17T^{2} \) |
| 19 | \( 1 + 7.09iT - 19T^{2} \) |
| 29 | \( 1 + 7.02iT - 29T^{2} \) |
| 31 | \( 1 - 2.54T + 31T^{2} \) |
| 37 | \( 1 + 11.8iT - 37T^{2} \) |
| 41 | \( 1 + 1.34iT - 41T^{2} \) |
| 43 | \( 1 - 7.71iT - 43T^{2} \) |
| 47 | \( 1 + 2.64iT - 47T^{2} \) |
| 53 | \( 1 + 5.21T + 53T^{2} \) |
| 59 | \( 1 - 9.91iT - 59T^{2} \) |
| 61 | \( 1 - 7.34iT - 61T^{2} \) |
| 67 | \( 1 + 1.63iT - 67T^{2} \) |
| 71 | \( 1 - 13.2iT - 71T^{2} \) |
| 73 | \( 1 + 10.0T + 73T^{2} \) |
| 79 | \( 1 - 0.0188iT - 79T^{2} \) |
| 83 | \( 1 + 1.02T + 83T^{2} \) |
| 89 | \( 1 - 8.49T + 89T^{2} \) |
| 97 | \( 1 + 12.5iT - 97T^{2} \) |
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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
−7.43613244864510506799599326643, −7.11200855416169562584567144615, −6.02590633248267467721839992340, −5.53405332028860681344129310162, −4.52848982514008961586198262111, −4.04591400463718561709167494443, −3.17749371567390299820658087109, −2.41449254150863562946782141760, −1.14891521029808481672064945478, −0.25180671237432304505926207074,
1.20223291951995248385080572054, 2.15620391730952091073203609004, 3.16550252159698166024423999196, 3.62727600546867799757049291037, 4.71956047077133017974032363787, 5.29185289063058645192322612481, 6.08160445836699647319727982907, 6.55102788431675691855465083204, 7.65111298342315820850153788298, 8.111949081692438426795019388441