| L(s) = 1 | − 3.32·3-s − 1.70·5-s + 8.02·9-s + 11-s + 3.32·13-s + 5.66·15-s − 0.867·17-s + 2.27·19-s − 2.61·23-s − 2.09·25-s − 16.6·27-s + 9.34·29-s − 8.59·31-s − 3.32·33-s − 7.41·37-s − 11.0·39-s − 4.83·41-s + 6.32·43-s − 13.6·45-s − 2.40·47-s + 2.88·51-s + 7.98·53-s − 1.70·55-s − 7.56·57-s − 11.0·59-s + 7.43·61-s − 5.66·65-s + ⋯ |
| L(s) = 1 | − 1.91·3-s − 0.762·5-s + 2.67·9-s + 0.301·11-s + 0.920·13-s + 1.46·15-s − 0.210·17-s + 0.522·19-s − 0.545·23-s − 0.418·25-s − 3.21·27-s + 1.73·29-s − 1.54·31-s − 0.578·33-s − 1.21·37-s − 1.76·39-s − 0.755·41-s + 0.963·43-s − 2.04·45-s − 0.351·47-s + 0.403·51-s + 1.09·53-s − 0.229·55-s − 1.00·57-s − 1.44·59-s + 0.951·61-s − 0.702·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 3.32T + 3T^{2} \) |
| 5 | \( 1 + 1.70T + 5T^{2} \) |
| 13 | \( 1 - 3.32T + 13T^{2} \) |
| 17 | \( 1 + 0.867T + 17T^{2} \) |
| 19 | \( 1 - 2.27T + 19T^{2} \) |
| 23 | \( 1 + 2.61T + 23T^{2} \) |
| 29 | \( 1 - 9.34T + 29T^{2} \) |
| 31 | \( 1 + 8.59T + 31T^{2} \) |
| 37 | \( 1 + 7.41T + 37T^{2} \) |
| 41 | \( 1 + 4.83T + 41T^{2} \) |
| 43 | \( 1 - 6.32T + 43T^{2} \) |
| 47 | \( 1 + 2.40T + 47T^{2} \) |
| 53 | \( 1 - 7.98T + 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 - 7.43T + 61T^{2} \) |
| 67 | \( 1 + 6.48T + 67T^{2} \) |
| 71 | \( 1 + 0.867T + 71T^{2} \) |
| 73 | \( 1 - 13.6T + 73T^{2} \) |
| 79 | \( 1 + 9.47T + 79T^{2} \) |
| 83 | \( 1 - 6.54T + 83T^{2} \) |
| 89 | \( 1 + 8.86T + 89T^{2} \) |
| 97 | \( 1 + 13.2T + 97T^{2} \) |
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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
−7.14339396589176547445427851600, −6.73816679085524811320238895186, −5.95400608598401028758693933571, −5.49345180373497918340882924343, −4.67943688991793218258627460274, −4.06469410841105378509870904284, −3.41414409086139950202212827462, −1.82418655663381761139873276203, −0.950294903072132692665701512843, 0,
0.950294903072132692665701512843, 1.82418655663381761139873276203, 3.41414409086139950202212827462, 4.06469410841105378509870904284, 4.67943688991793218258627460274, 5.49345180373497918340882924343, 5.95400608598401028758693933571, 6.73816679085524811320238895186, 7.14339396589176547445427851600
