L(s) = 1 | − 5·5-s + 27.5·7-s − 40.6·11-s − 67.8·13-s − 91.1·17-s + 37.9·19-s − 57.7·23-s + 25·25-s + 49.8·29-s + 226.·31-s − 137.·35-s − 144.·37-s + 348.·41-s + 455.·43-s + 179.·47-s + 418.·49-s − 300.·53-s + 203.·55-s − 439.·59-s − 191.·61-s + 339.·65-s + 462.·67-s + 671.·71-s − 315.·73-s − 1.12e3·77-s + 1.00e3·79-s + 1.20e3·83-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 1.48·7-s − 1.11·11-s − 1.44·13-s − 1.29·17-s + 0.457·19-s − 0.523·23-s + 0.200·25-s + 0.319·29-s + 1.30·31-s − 0.666·35-s − 0.640·37-s + 1.32·41-s + 1.61·43-s + 0.556·47-s + 1.21·49-s − 0.779·53-s + 0.498·55-s − 0.970·59-s − 0.402·61-s + 0.647·65-s + 0.843·67-s + 1.12·71-s − 0.506·73-s − 1.65·77-s + 1.43·79-s + 1.59·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.685571772\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.685571772\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
good | 7 | \( 1 - 27.5T + 343T^{2} \) |
| 11 | \( 1 + 40.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 67.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 91.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 37.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 57.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 49.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 226.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 144.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 348.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 455.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 179.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 300.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 439.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 191.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 462.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 671.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 315.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.00e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.20e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 981.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.84e3T + 9.12e5T^{2} \) |
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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
−8.942963777612603846325140185925, −7.88035972290058235255925239115, −7.81501370757524515042672825284, −6.78341517366664482641487384202, −5.55325774413622143849596446167, −4.76196750944738910989426328676, −4.32532608457883809219490899534, −2.74581869551032436106933433282, −2.06440495359369255977754401300, −0.60679390505036620557956778631,
0.60679390505036620557956778631, 2.06440495359369255977754401300, 2.74581869551032436106933433282, 4.32532608457883809219490899534, 4.76196750944738910989426328676, 5.55325774413622143849596446167, 6.78341517366664482641487384202, 7.81501370757524515042672825284, 7.88035972290058235255925239115, 8.942963777612603846325140185925