| L(s) = 1 | − 450·3-s − 1.07e4·5-s − 2.14e4·7-s + 2.53e4·9-s + 1.59e5·11-s + 2.43e6·13-s + 4.82e6·15-s − 6.18e6·17-s − 1.31e7·19-s + 9.64e6·21-s + 2.89e7·23-s + 6.60e7·25-s + 6.83e7·27-s − 7.52e7·29-s − 8.78e7·31-s − 7.17e7·33-s + 2.29e8·35-s − 4.69e8·37-s − 1.09e9·39-s − 4.48e8·41-s − 1.58e9·43-s − 2.71e8·45-s − 2.89e9·47-s − 1.51e9·49-s + 2.78e9·51-s + 2.53e9·53-s − 1.70e9·55-s + ⋯ |
| L(s) = 1 | − 1.06·3-s − 1.53·5-s − 0.482·7-s + 0.143·9-s + 0.298·11-s + 1.81·13-s + 1.63·15-s − 1.05·17-s − 1.21·19-s + 0.515·21-s + 0.936·23-s + 1.35·25-s + 0.916·27-s − 0.681·29-s − 0.551·31-s − 0.319·33-s + 0.739·35-s − 1.11·37-s − 1.94·39-s − 0.604·41-s − 1.64·43-s − 0.219·45-s − 1.84·47-s − 0.767·49-s + 1.13·51-s + 0.833·53-s − 0.457·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.09398367369\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09398367369\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 450T + 1.77e5T^{2} \) |
| 5 | \( 1 + 1.07e4T + 4.88e7T^{2} \) |
| 7 | \( 1 + 2.14e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 1.59e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 2.43e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + 6.18e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 1.31e7T + 1.16e14T^{2} \) |
| 23 | \( 1 - 2.89e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 7.52e7T + 1.22e16T^{2} \) |
| 31 | \( 1 + 8.78e7T + 2.54e16T^{2} \) |
| 37 | \( 1 + 4.69e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 4.48e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.58e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + 2.89e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 2.53e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + 2.23e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 1.61e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 7.00e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + 1.98e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 8.21e9T + 3.13e20T^{2} \) |
| 79 | \( 1 + 3.76e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 5.92e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 5.32e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 9.64e10T + 7.15e21T^{2} \) |
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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
−10.51848759143142755221531414112, −8.847836711662485063661641631808, −8.316154500150022736547104663884, −6.84135921046674616950807118262, −6.38442197080959004265227389036, −5.08671847820636067343274967734, −4.02424850828909695486724847200, −3.29415506683587042369313918307, −1.44694273136514278235913616323, −0.14255956118450369051984505941,
0.14255956118450369051984505941, 1.44694273136514278235913616323, 3.29415506683587042369313918307, 4.02424850828909695486724847200, 5.08671847820636067343274967734, 6.38442197080959004265227389036, 6.84135921046674616950807118262, 8.316154500150022736547104663884, 8.847836711662485063661641631808, 10.51848759143142755221531414112
