| L(s) = 1 | − 19.1·3-s + 9·5-s + 157.·7-s + 124.·9-s + 121·11-s − 317.·13-s − 172.·15-s − 2.23e3·17-s + 649.·19-s − 3.02e3·21-s − 2.23e3·23-s − 3.04e3·25-s + 2.27e3·27-s − 883.·29-s − 7.03e3·31-s − 2.31e3·33-s + 1.42e3·35-s + 2.77e3·37-s + 6.08e3·39-s − 807.·41-s + 4.10e3·43-s + 1.11e3·45-s − 1.54e4·47-s + 8.10e3·49-s + 4.27e4·51-s − 2.84e4·53-s + 1.08e3·55-s + ⋯ |
| L(s) = 1 | − 1.22·3-s + 0.160·5-s + 1.21·7-s + 0.511·9-s + 0.301·11-s − 0.521·13-s − 0.197·15-s − 1.87·17-s + 0.412·19-s − 1.49·21-s − 0.881·23-s − 0.974·25-s + 0.600·27-s − 0.195·29-s − 1.31·31-s − 0.370·33-s + 0.195·35-s + 0.332·37-s + 0.640·39-s − 0.0750·41-s + 0.338·43-s + 0.0823·45-s − 1.01·47-s + 0.481·49-s + 2.30·51-s − 1.39·53-s + 0.0485·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - 121T \) |
| good | 3 | \( 1 + 19.1T + 243T^{2} \) |
| 5 | \( 1 - 9T + 3.12e3T^{2} \) |
| 7 | \( 1 - 157.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 317.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 2.23e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 649.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.23e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 883.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.03e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.77e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 807.T + 1.15e8T^{2} \) |
| 43 | \( 1 - 4.10e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.54e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.84e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.76e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.53e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.29e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.48e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.98e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.12e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.39e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.96e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.23e5T + 8.58e9T^{2} \) |
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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
−12.38533417131982604553854871315, −11.39316500066071545222518069164, −10.91933652588052329377717055103, −9.437600178168344398852270225392, −8.009116904996134824652630165094, −6.63005373551509837687787110993, −5.43546242372640507626869365975, −4.40451790808661128740963519756, −1.86402346381190945832539912020, 0,
1.86402346381190945832539912020, 4.40451790808661128740963519756, 5.43546242372640507626869365975, 6.63005373551509837687787110993, 8.009116904996134824652630165094, 9.437600178168344398852270225392, 10.91933652588052329377717055103, 11.39316500066071545222518069164, 12.38533417131982604553854871315
