| L(s) = 1 | + 2.15·5-s + 0.571·7-s + 11-s − 13-s + 2.82·17-s − 8.00·19-s − 4.07·23-s − 0.337·25-s − 4.46·29-s − 8.10·31-s + 1.23·35-s − 2.75·37-s − 12.5·41-s + 10.4·43-s − 3.60·47-s − 6.67·49-s + 4.57·53-s + 2.15·55-s − 10.7·59-s + 0.963·61-s − 2.15·65-s + 1.35·67-s − 7.56·71-s − 6.06·73-s + 0.571·77-s − 5.75·79-s − 8.01·83-s + ⋯ |
| L(s) = 1 | + 0.965·5-s + 0.216·7-s + 0.301·11-s − 0.277·13-s + 0.684·17-s − 1.83·19-s − 0.850·23-s − 0.0674·25-s − 0.828·29-s − 1.45·31-s + 0.208·35-s − 0.452·37-s − 1.96·41-s + 1.60·43-s − 0.526·47-s − 0.953·49-s + 0.628·53-s + 0.291·55-s − 1.40·59-s + 0.123·61-s − 0.267·65-s + 0.165·67-s − 0.897·71-s − 0.709·73-s + 0.0651·77-s − 0.647·79-s − 0.880·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 2.15T + 5T^{2} \) |
| 7 | \( 1 - 0.571T + 7T^{2} \) |
| 17 | \( 1 - 2.82T + 17T^{2} \) |
| 19 | \( 1 + 8.00T + 19T^{2} \) |
| 23 | \( 1 + 4.07T + 23T^{2} \) |
| 29 | \( 1 + 4.46T + 29T^{2} \) |
| 31 | \( 1 + 8.10T + 31T^{2} \) |
| 37 | \( 1 + 2.75T + 37T^{2} \) |
| 41 | \( 1 + 12.5T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 + 3.60T + 47T^{2} \) |
| 53 | \( 1 - 4.57T + 53T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 61 | \( 1 - 0.963T + 61T^{2} \) |
| 67 | \( 1 - 1.35T + 67T^{2} \) |
| 71 | \( 1 + 7.56T + 71T^{2} \) |
| 73 | \( 1 + 6.06T + 73T^{2} \) |
| 79 | \( 1 + 5.75T + 79T^{2} \) |
| 83 | \( 1 + 8.01T + 83T^{2} \) |
| 89 | \( 1 - 5.50T + 89T^{2} \) |
| 97 | \( 1 - 14.4T + 97T^{2} \) |
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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
−7.82480190587628873545811751136, −7.15443890909702402865319375205, −6.22140211265120813768335168132, −5.83947426660101885467556052596, −4.99472512680650144747295642242, −4.14281873854951622335762077093, −3.31126271491695600647361239077, −2.07776500234413590868044277658, −1.69854002718027246866305192807, 0,
1.69854002718027246866305192807, 2.07776500234413590868044277658, 3.31126271491695600647361239077, 4.14281873854951622335762077093, 4.99472512680650144747295642242, 5.83947426660101885467556052596, 6.22140211265120813768335168132, 7.15443890909702402865319375205, 7.82480190587628873545811751136
