| L(s) = 1 | − 7.89·2-s − 81·3-s − 449.·4-s + 876.·5-s + 639.·6-s − 1.20e3·7-s + 7.59e3·8-s + 6.56e3·9-s − 6.92e3·10-s + 3.88e4·11-s + 3.64e4·12-s + 1.83e5·13-s + 9.49e3·14-s − 7.09e4·15-s + 1.70e5·16-s + 1.39e5·17-s − 5.18e4·18-s − 8.67e5·19-s − 3.94e5·20-s + 9.73e4·21-s − 3.06e5·22-s + 1.16e6·23-s − 6.15e5·24-s − 1.18e6·25-s − 1.44e6·26-s − 5.31e5·27-s + 5.40e5·28-s + ⋯ |
| L(s) = 1 | − 0.348·2-s − 0.577·3-s − 0.878·4-s + 0.627·5-s + 0.201·6-s − 0.189·7-s + 0.655·8-s + 0.333·9-s − 0.218·10-s + 0.799·11-s + 0.507·12-s + 1.78·13-s + 0.0660·14-s − 0.362·15-s + 0.649·16-s + 0.405·17-s − 0.116·18-s − 1.52·19-s − 0.550·20-s + 0.109·21-s − 0.278·22-s + 0.869·23-s − 0.378·24-s − 0.606·25-s − 0.621·26-s − 0.192·27-s + 0.166·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.463380594\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.463380594\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 81T \) |
| 59 | \( 1 - 1.21e7T \) |
| good | 2 | \( 1 + 7.89T + 512T^{2} \) |
| 5 | \( 1 - 876.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 1.20e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 3.88e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.83e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 1.39e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 8.67e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.16e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 2.91e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 4.99e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.18e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.41e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.70e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.28e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 7.27e7T + 3.29e15T^{2} \) |
| 61 | \( 1 + 9.05e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.29e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.92e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 4.07e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.64e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.68e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 6.83e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 5.92e8T + 7.60e17T^{2} \) |
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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
−10.78241768522433898036051660312, −10.04354303182917085463377591419, −8.983434175726170635087222979404, −8.294511395944761697319553954929, −6.62801009055504783207215745109, −5.89092251602079568231456769345, −4.61554331369367529824318416437, −3.58188983148670416638293689040, −1.60970473404038771787485653126, −0.70819950617079958604687574660,
0.70819950617079958604687574660, 1.60970473404038771787485653126, 3.58188983148670416638293689040, 4.61554331369367529824318416437, 5.89092251602079568231456769345, 6.62801009055504783207215745109, 8.294511395944761697319553954929, 8.983434175726170635087222979404, 10.04354303182917085463377591419, 10.78241768522433898036051660312
