| L(s) = 1 | − 3-s − 5-s − 4·7-s + 9-s − 13-s + 15-s − 3·19-s + 4·21-s − 2·23-s + 25-s − 27-s + 3·29-s − 4·31-s + 4·35-s + 10·37-s + 39-s − 5·41-s + 8·43-s − 45-s + 7·47-s + 9·49-s − 3·53-s + 3·57-s + 8·61-s − 4·63-s + 65-s − 5·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s − 0.277·13-s + 0.258·15-s − 0.688·19-s + 0.872·21-s − 0.417·23-s + 1/5·25-s − 0.192·27-s + 0.557·29-s − 0.718·31-s + 0.676·35-s + 1.64·37-s + 0.160·39-s − 0.780·41-s + 1.21·43-s − 0.149·45-s + 1.02·47-s + 9/7·49-s − 0.412·53-s + 0.397·57-s + 1.02·61-s − 0.503·63-s + 0.124·65-s − 0.610·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.170981100\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.170981100\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−12.92567658896995, −12.52186300287602, −12.40685241343473, −11.74378915473576, −11.17130220550472, −10.87971153240516, −10.13244381805383, −9.976706462252908, −9.388838033386341, −8.913623802295160, −8.396848194361037, −7.685445215388785, −7.328098888592263, −6.752848362257025, −6.307987130598510, −5.938238085466594, −5.400902200184555, −4.647456363360872, −4.205979523261016, −3.688715456838046, −3.121890883039771, −2.510928106637383, −1.920923291982412, −0.8169120110626342, −0.4313217308486578,
0.4313217308486578, 0.8169120110626342, 1.920923291982412, 2.510928106637383, 3.121890883039771, 3.688715456838046, 4.205979523261016, 4.647456363360872, 5.400902200184555, 5.938238085466594, 6.307987130598510, 6.752848362257025, 7.328098888592263, 7.685445215388785, 8.396848194361037, 8.913623802295160, 9.388838033386341, 9.976706462252908, 10.13244381805383, 10.87971153240516, 11.17130220550472, 11.74378915473576, 12.40685241343473, 12.52186300287602, 12.92567658896995
