| L(s) = 1 | + 176.·3-s + 1.80e3·5-s + 387.·7-s + 1.14e4·9-s + 7.55e4·11-s + 2.85e4·13-s + 3.17e5·15-s + 3.09e5·17-s − 3.67e5·19-s + 6.83e4·21-s − 6.44e4·23-s + 1.29e6·25-s − 1.45e6·27-s + 2.76e6·29-s + 1.47e6·31-s + 1.33e7·33-s + 6.97e5·35-s − 1.16e6·37-s + 5.03e6·39-s + 3.80e6·41-s + 2.53e7·43-s + 2.06e7·45-s + 2.93e7·47-s − 4.02e7·49-s + 5.45e7·51-s − 7.12e7·53-s + 1.36e8·55-s + ⋯ |
| L(s) = 1 | + 1.25·3-s + 1.28·5-s + 0.0609·7-s + 0.581·9-s + 1.55·11-s + 0.277·13-s + 1.62·15-s + 0.898·17-s − 0.647·19-s + 0.0766·21-s − 0.0479·23-s + 0.661·25-s − 0.526·27-s + 0.726·29-s + 0.287·31-s + 1.95·33-s + 0.0786·35-s − 0.101·37-s + 0.348·39-s + 0.210·41-s + 1.12·43-s + 0.749·45-s + 0.878·47-s − 0.996·49-s + 1.12·51-s − 1.24·53-s + 2.00·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(5.519515604\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.519515604\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - 2.85e4T \) |
| good | 3 | \( 1 - 176.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.80e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 387.T + 4.03e7T^{2} \) |
| 11 | \( 1 - 7.55e4T + 2.35e9T^{2} \) |
| 17 | \( 1 - 3.09e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 3.67e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 6.44e4T + 1.80e12T^{2} \) |
| 29 | \( 1 - 2.76e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 1.47e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.16e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 3.80e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.53e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.93e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 7.12e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 3.11e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.00e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 6.31e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.79e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.78e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.42e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 5.43e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 2.08e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 2.50e7T + 7.60e17T^{2} \) |
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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
−10.43543491503022066218020320841, −9.474004617389354891570876462333, −8.984687789547394855422944299675, −7.972427541781677083608738527383, −6.64031128409990674918143356249, −5.76956292317951199464146252911, −4.21351755944862401377473298911, −3.10730659873043153609009149819, −2.03375664544225356500983203018, −1.17078673870739115676480532859,
1.17078673870739115676480532859, 2.03375664544225356500983203018, 3.10730659873043153609009149819, 4.21351755944862401377473298911, 5.76956292317951199464146252911, 6.64031128409990674918143356249, 7.972427541781677083608738527383, 8.984687789547394855422944299675, 9.474004617389354891570876462333, 10.43543491503022066218020320841
