| L(s) = 1 | + 3-s + 7-s + 9-s − 11-s − 13-s + 17-s + 19-s + 21-s − 23-s + 25-s + 27-s − 33-s + 37-s − 39-s − 2·43-s + 51-s + 53-s + 57-s + 2·61-s + 63-s − 69-s − 73-s + 75-s − 77-s + 81-s − 83-s + 89-s + ⋯ |
| L(s) = 1 | + 3-s + 7-s + 9-s − 11-s − 13-s + 17-s + 19-s + 21-s − 23-s + 25-s + 27-s − 33-s + 37-s − 39-s − 2·43-s + 51-s + 53-s + 57-s + 2·61-s + 63-s − 69-s − 73-s + 75-s − 77-s + 81-s − 83-s + 89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.000122992\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.000122992\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| good | 5 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( 1 - T + T^{2} \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( 1 - T + T^{2} \) |
| 19 | \( 1 - T + T^{2} \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 + T )^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 - T + T^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( 1 + T + T^{2} \) |
| 89 | \( 1 - T + T^{2} \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−8.478944208485155164891560183216, −8.035058913203481345421992084829, −7.52338383795473679955320225120, −6.80833128791609150745454103920, −5.42784336591383814309920026956, −5.00970709267338656494472321849, −4.08141196160836239637394921546, −3.05753936816179205965472800862, −2.37691471345372383938296122960, −1.32851897056744297827539666402,
1.32851897056744297827539666402, 2.37691471345372383938296122960, 3.05753936816179205965472800862, 4.08141196160836239637394921546, 5.00970709267338656494472321849, 5.42784336591383814309920026956, 6.80833128791609150745454103920, 7.52338383795473679955320225120, 8.035058913203481345421992084829, 8.478944208485155164891560183216
