| L(s) = 1 | + 0.810·3-s + 71.6·5-s − 49·7-s − 242.·9-s + 569.·11-s + 137.·13-s + 58.0·15-s − 418.·17-s + 2.55e3·19-s − 39.7·21-s − 127.·23-s + 2.01e3·25-s − 393.·27-s + 2.31e3·29-s − 3.99e3·31-s + 461.·33-s − 3.51e3·35-s − 3.85e3·37-s + 111.·39-s − 4.94e3·41-s + 1.36e4·43-s − 1.73e4·45-s + 2.76e4·47-s + 2.40e3·49-s − 339.·51-s + 3.73e4·53-s + 4.08e4·55-s + ⋯ |
| L(s) = 1 | + 0.0519·3-s + 1.28·5-s − 0.377·7-s − 0.997·9-s + 1.41·11-s + 0.225·13-s + 0.0666·15-s − 0.351·17-s + 1.62·19-s − 0.0196·21-s − 0.0500·23-s + 0.643·25-s − 0.103·27-s + 0.510·29-s − 0.746·31-s + 0.0737·33-s − 0.484·35-s − 0.462·37-s + 0.0117·39-s − 0.459·41-s + 1.12·43-s − 1.27·45-s + 1.82·47-s + 0.142·49-s − 0.0182·51-s + 1.82·53-s + 1.81·55-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
| \(L(3)\) |
\(\approx\) |
\(2.630006414\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.630006414\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 0.810T + 243T^{2} \) |
| 5 | \( 1 - 71.6T + 3.12e3T^{2} \) |
| 11 | \( 1 - 569.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 137.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 418.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.55e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 127.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.31e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.99e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 3.85e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 4.94e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.36e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.76e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.73e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.69e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.80e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.24e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.50e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.92e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.09e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.97e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.04e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 9.66e4T + 8.58e9T^{2} \) |
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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
−11.45319670923299355643925732946, −10.27894901779748889729001180206, −9.304338076822741088434643129833, −8.811361051441980219871438239035, −7.17070242705292405579710771628, −6.11897209167785143791704398550, −5.41360389524704514500132330249, −3.70483769102511447093288448136, −2.40804573130240669842264132597, −1.02793582829200226071371025714,
1.02793582829200226071371025714, 2.40804573130240669842264132597, 3.70483769102511447093288448136, 5.41360389524704514500132330249, 6.11897209167785143791704398550, 7.17070242705292405579710771628, 8.811361051441980219871438239035, 9.304338076822741088434643129833, 10.27894901779748889729001180206, 11.45319670923299355643925732946
