| L(s) = 1 | − 14.8·3-s − 37.6·5-s − 49·7-s − 23.6·9-s − 149.·11-s − 627.·13-s + 557.·15-s − 637.·17-s − 1.30e3·19-s + 725.·21-s − 376.·23-s − 1.70e3·25-s + 3.94e3·27-s − 6.21e3·29-s + 6.03e3·31-s + 2.21e3·33-s + 1.84e3·35-s − 3.63e3·37-s + 9.29e3·39-s + 1.27e4·41-s − 3.34e3·43-s + 891.·45-s + 1.43e4·47-s + 2.40e3·49-s + 9.43e3·51-s − 4.58e3·53-s + 5.62e3·55-s + ⋯ |
| L(s) = 1 | − 0.950·3-s − 0.673·5-s − 0.377·7-s − 0.0973·9-s − 0.371·11-s − 1.03·13-s + 0.640·15-s − 0.534·17-s − 0.830·19-s + 0.359·21-s − 0.148·23-s − 0.545·25-s + 1.04·27-s − 1.37·29-s + 1.12·31-s + 0.353·33-s + 0.254·35-s − 0.436·37-s + 0.978·39-s + 1.18·41-s − 0.276·43-s + 0.0656·45-s + 0.947·47-s + 0.142·49-s + 0.508·51-s − 0.224·53-s + 0.250·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.4141534348\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4141534348\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + 49T \) |
| good | 3 | \( 1 + 14.8T + 243T^{2} \) |
| 5 | \( 1 + 37.6T + 3.12e3T^{2} \) |
| 11 | \( 1 + 149.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 627.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 637.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.30e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 376.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.21e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.03e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 3.63e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.27e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 3.34e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.43e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 4.58e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.13e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.08e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.52e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.89e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.51e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.36e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.80e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.21e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 646.T + 8.58e9T^{2} \) |
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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
−11.41276710441196694916770207739, −10.63244333028578195602037247401, −9.569369161247496113401394444188, −8.318899304455427856038960175580, −7.25635983127309522821361071813, −6.21307047659836030599724815678, −5.15925342411593564007758354475, −4.03358900424641313592678229534, −2.45598976059752256448345259672, −0.38130564166366943337714664698,
0.38130564166366943337714664698, 2.45598976059752256448345259672, 4.03358900424641313592678229534, 5.15925342411593564007758354475, 6.21307047659836030599724815678, 7.25635983127309522821361071813, 8.318899304455427856038960175580, 9.569369161247496113401394444188, 10.63244333028578195602037247401, 11.41276710441196694916770207739
