| L(s) = 1 | + 3-s + 4·5-s − 2·9-s − 3·11-s + 2·13-s + 4·15-s + 2·17-s + 6·23-s + 11·25-s − 5·27-s − 4·29-s + 10·31-s − 3·33-s + 2·37-s + 2·39-s − 9·41-s + 4·43-s − 8·45-s − 12·47-s − 7·49-s + 2·51-s − 2·53-s − 12·55-s − 59-s + 8·61-s + 8·65-s + 9·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.78·5-s − 2/3·9-s − 0.904·11-s + 0.554·13-s + 1.03·15-s + 0.485·17-s + 1.25·23-s + 11/5·25-s − 0.962·27-s − 0.742·29-s + 1.79·31-s − 0.522·33-s + 0.328·37-s + 0.320·39-s − 1.40·41-s + 0.609·43-s − 1.19·45-s − 1.75·47-s − 49-s + 0.280·51-s − 0.274·53-s − 1.61·55-s − 0.130·59-s + 1.02·61-s + 0.992·65-s + 1.09·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.076575828\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.076575828\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 5 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−15.35946606552246, −14.67634121799579, −14.46792818754077, −13.72128475803526, −13.34452395392398, −13.12621291614501, −12.42106305900373, −11.52060573944077, −11.05257576599346, −10.40857287252990, −9.827870380114998, −9.532053653258567, −8.797908675595239, −8.349162064580453, −7.807983298022787, −6.868810600507119, −6.324232007364210, −5.799020080838438, −5.143140298048136, −4.790792960577682, −3.419301022590749, −3.031584456782738, −2.327726819952727, −1.729222322341244, −0.7823831571132692,
0.7823831571132692, 1.729222322341244, 2.327726819952727, 3.031584456782738, 3.419301022590749, 4.790792960577682, 5.143140298048136, 5.799020080838438, 6.324232007364210, 6.868810600507119, 7.807983298022787, 8.349162064580453, 8.797908675595239, 9.532053653258567, 9.827870380114998, 10.40857287252990, 11.05257576599346, 11.52060573944077, 12.42106305900373, 13.12621291614501, 13.34452395392398, 13.72128475803526, 14.46792818754077, 14.67634121799579, 15.35946606552246
