| L(s) = 1 | + 3.31·3-s + 3.31·7-s + 8·9-s + 3.31·11-s − 3·17-s + 3.31·19-s + 11·21-s − 3.31·23-s − 5·25-s + 16.5·27-s − 5·29-s + 11·33-s + 9·37-s − 3·41-s + 9.94·43-s − 6.63·47-s + 4·49-s − 9.94·51-s − 8·53-s + 11·57-s + 3.31·59-s − 9·61-s + 26.5·63-s + 9.94·67-s − 11·69-s + 9.94·71-s − 4·73-s + ⋯ |
| L(s) = 1 | + 1.91·3-s + 1.25·7-s + 2.66·9-s + 1.00·11-s − 0.727·17-s + 0.760·19-s + 2.40·21-s − 0.691·23-s − 25-s + 3.19·27-s − 0.928·29-s + 1.91·33-s + 1.47·37-s − 0.468·41-s + 1.51·43-s − 0.967·47-s + 0.571·49-s − 1.39·51-s − 1.09·53-s + 1.45·57-s + 0.431·59-s − 1.15·61-s + 3.34·63-s + 1.21·67-s − 1.32·69-s + 1.18·71-s − 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.357688750\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.357688750\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 3.31T + 3T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 - 3.31T + 7T^{2} \) |
| 11 | \( 1 - 3.31T + 11T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 - 3.31T + 19T^{2} \) |
| 23 | \( 1 + 3.31T + 23T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 9T + 37T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 - 9.94T + 43T^{2} \) |
| 47 | \( 1 + 6.63T + 47T^{2} \) |
| 53 | \( 1 + 8T + 53T^{2} \) |
| 59 | \( 1 - 3.31T + 59T^{2} \) |
| 61 | \( 1 + 9T + 61T^{2} \) |
| 67 | \( 1 - 9.94T + 67T^{2} \) |
| 71 | \( 1 - 9.94T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 - 6.63T + 79T^{2} \) |
| 83 | \( 1 + 13.2T + 83T^{2} \) |
| 89 | \( 1 + T + 89T^{2} \) |
| 97 | \( 1 - 7T + 97T^{2} \) |
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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
−8.028311944909076092250368420788, −7.82933715641129474241412091755, −7.05399469174773880524133072779, −6.16507443014247462017687500131, −5.00113114810280731781938136005, −4.15341009478844214856371815292, −3.81000083226893259765993328479, −2.72914034810044166615713316978, −1.93633525696591744682747206544, −1.33093685204481500899422458147,
1.33093685204481500899422458147, 1.93633525696591744682747206544, 2.72914034810044166615713316978, 3.81000083226893259765993328479, 4.15341009478844214856371815292, 5.00113114810280731781938136005, 6.16507443014247462017687500131, 7.05399469174773880524133072779, 7.82933715641129474241412091755, 8.028311944909076092250368420788
