| L(s) = 1 | + 9·3-s + 28.9·5-s − 210.·7-s + 81·9-s − 169.·11-s − 1.01e3·13-s + 260.·15-s − 1.84e3·17-s − 361·19-s − 1.89e3·21-s + 1.41e3·23-s − 2.28e3·25-s + 729·27-s + 2.79e3·29-s − 61.3·31-s − 1.52e3·33-s − 6.10e3·35-s + 2.24e3·37-s − 9.17e3·39-s + 997.·41-s + 3.73e3·43-s + 2.34e3·45-s + 1.60e4·47-s + 2.76e4·49-s − 1.66e4·51-s + 3.95e4·53-s − 4.89e3·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.517·5-s − 1.62·7-s + 0.333·9-s − 0.421·11-s − 1.67·13-s + 0.298·15-s − 1.55·17-s − 0.229·19-s − 0.939·21-s + 0.558·23-s − 0.732·25-s + 0.192·27-s + 0.618·29-s − 0.0114·31-s − 0.243·33-s − 0.841·35-s + 0.269·37-s − 0.966·39-s + 0.0926·41-s + 0.307·43-s + 0.172·45-s + 1.06·47-s + 1.64·49-s − 0.895·51-s + 1.93·53-s − 0.218·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.344105005\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.344105005\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 9T \) |
| 19 | \( 1 + 361T \) |
| good | 5 | \( 1 - 28.9T + 3.12e3T^{2} \) |
| 7 | \( 1 + 210.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 169.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.01e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.84e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 1.41e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.79e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 61.3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.24e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 997.T + 1.15e8T^{2} \) |
| 43 | \( 1 - 3.73e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.60e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.95e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.54e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.62e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.65e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 8.88e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.25e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.32e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.72e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.69e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 4.61e4T + 8.58e9T^{2} \) |
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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
−9.383527547174066375509503701658, −8.779866835168843056191706202314, −7.50880117963519390919098533148, −6.85170339466552708098558352272, −6.04971793420649790704662761788, −4.91735873323281062044175487675, −3.88902481471202629630378833329, −2.66703894874464617301262893119, −2.29417454733663311066615778472, −0.46291247749382633284912468113,
0.46291247749382633284912468113, 2.29417454733663311066615778472, 2.66703894874464617301262893119, 3.88902481471202629630378833329, 4.91735873323281062044175487675, 6.04971793420649790704662761788, 6.85170339466552708098558352272, 7.50880117963519390919098533148, 8.779866835168843056191706202314, 9.383527547174066375509503701658
