| L(s) = 1 | − 1.17·3-s + 2.17·5-s + 7-s − 1.61·9-s + 2.52·11-s − 4.02·13-s − 2.55·15-s + 17-s − 5.53·19-s − 1.17·21-s − 4.07·23-s − 0.266·25-s + 5.42·27-s + 4.42·29-s + 2.37·31-s − 2.96·33-s + 2.17·35-s + 6.64·37-s + 4.73·39-s + 7.85·41-s + 9.17·43-s − 3.52·45-s − 12.6·47-s + 49-s − 1.17·51-s + 3.95·53-s + 5.48·55-s + ⋯ |
| L(s) = 1 | − 0.678·3-s + 0.972·5-s + 0.377·7-s − 0.539·9-s + 0.759·11-s − 1.11·13-s − 0.660·15-s + 0.242·17-s − 1.26·19-s − 0.256·21-s − 0.850·23-s − 0.0533·25-s + 1.04·27-s + 0.821·29-s + 0.426·31-s − 0.515·33-s + 0.367·35-s + 1.09·37-s + 0.758·39-s + 1.22·41-s + 1.39·43-s − 0.524·45-s − 1.84·47-s + 0.142·49-s − 0.164·51-s + 0.542·53-s + 0.739·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + 1.17T + 3T^{2} \) |
| 5 | \( 1 - 2.17T + 5T^{2} \) |
| 11 | \( 1 - 2.52T + 11T^{2} \) |
| 13 | \( 1 + 4.02T + 13T^{2} \) |
| 19 | \( 1 + 5.53T + 19T^{2} \) |
| 23 | \( 1 + 4.07T + 23T^{2} \) |
| 29 | \( 1 - 4.42T + 29T^{2} \) |
| 31 | \( 1 - 2.37T + 31T^{2} \) |
| 37 | \( 1 - 6.64T + 37T^{2} \) |
| 41 | \( 1 - 7.85T + 41T^{2} \) |
| 43 | \( 1 - 9.17T + 43T^{2} \) |
| 47 | \( 1 + 12.6T + 47T^{2} \) |
| 53 | \( 1 - 3.95T + 53T^{2} \) |
| 59 | \( 1 + 14.9T + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 + 2.98T + 67T^{2} \) |
| 71 | \( 1 + 7.86T + 71T^{2} \) |
| 73 | \( 1 + 2.32T + 73T^{2} \) |
| 79 | \( 1 + 1.83T + 79T^{2} \) |
| 83 | \( 1 - 15.0T + 83T^{2} \) |
| 89 | \( 1 - 0.541T + 89T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
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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
−7.59233405854473955938743195176, −6.39473336835752220878288911460, −6.26812127264973593819251874127, −5.56963743894999302515653257630, −4.69808786375592453126089706171, −4.24358712490388287300353561889, −2.88687067534941586952659281812, −2.23512973828654404504520326338, −1.28149416781536503721554254703, 0,
1.28149416781536503721554254703, 2.23512973828654404504520326338, 2.88687067534941586952659281812, 4.24358712490388287300353561889, 4.69808786375592453126089706171, 5.56963743894999302515653257630, 6.26812127264973593819251874127, 6.39473336835752220878288911460, 7.59233405854473955938743195176
