| L(s) = 1 | − 2-s − 0.898·3-s + 4-s + 0.684·5-s + 0.898·6-s − 3.14·7-s − 8-s − 2.19·9-s − 0.684·10-s + 1.39·11-s − 0.898·12-s + 3.14·14-s − 0.614·15-s + 16-s − 1.95·17-s + 2.19·18-s + 19-s + 0.684·20-s + 2.82·21-s − 1.39·22-s + 2.21·23-s + 0.898·24-s − 4.53·25-s + 4.66·27-s − 3.14·28-s + 3.85·29-s + 0.614·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.518·3-s + 0.5·4-s + 0.305·5-s + 0.366·6-s − 1.18·7-s − 0.353·8-s − 0.730·9-s − 0.216·10-s + 0.421·11-s − 0.259·12-s + 0.839·14-s − 0.158·15-s + 0.250·16-s − 0.473·17-s + 0.516·18-s + 0.229·19-s + 0.152·20-s + 0.616·21-s − 0.298·22-s + 0.462·23-s + 0.183·24-s − 0.906·25-s + 0.897·27-s − 0.593·28-s + 0.715·29-s + 0.112·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 0.898T + 3T^{2} \) |
| 5 | \( 1 - 0.684T + 5T^{2} \) |
| 7 | \( 1 + 3.14T + 7T^{2} \) |
| 11 | \( 1 - 1.39T + 11T^{2} \) |
| 17 | \( 1 + 1.95T + 17T^{2} \) |
| 23 | \( 1 - 2.21T + 23T^{2} \) |
| 29 | \( 1 - 3.85T + 29T^{2} \) |
| 31 | \( 1 - 1.71T + 31T^{2} \) |
| 37 | \( 1 - 2.52T + 37T^{2} \) |
| 41 | \( 1 - 6.44T + 41T^{2} \) |
| 43 | \( 1 + 2.64T + 43T^{2} \) |
| 47 | \( 1 + 1.10T + 47T^{2} \) |
| 53 | \( 1 + 9.51T + 53T^{2} \) |
| 59 | \( 1 - 4.41T + 59T^{2} \) |
| 61 | \( 1 - 12.3T + 61T^{2} \) |
| 67 | \( 1 - 7.06T + 67T^{2} \) |
| 71 | \( 1 + 8.27T + 71T^{2} \) |
| 73 | \( 1 - 11.1T + 73T^{2} \) |
| 79 | \( 1 + 5.19T + 79T^{2} \) |
| 83 | \( 1 + 1.27T + 83T^{2} \) |
| 89 | \( 1 + 0.674T + 89T^{2} \) |
| 97 | \( 1 - 5.68T + 97T^{2} \) |
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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
−7.70605800688565075421794447506, −6.75070990543201600402007701187, −6.40002045637987059341649216792, −5.79888785569860473566122197574, −4.97774300860359390667788086735, −3.87890502054988798096856066140, −3.02164441052423059993825556752, −2.30740487923882923907172766190, −1.00817903991399240560729964430, 0,
1.00817903991399240560729964430, 2.30740487923882923907172766190, 3.02164441052423059993825556752, 3.87890502054988798096856066140, 4.97774300860359390667788086735, 5.79888785569860473566122197574, 6.40002045637987059341649216792, 6.75070990543201600402007701187, 7.70605800688565075421794447506
