| L(s) = 1 | − 2.32·2-s − 3·3-s − 2.57·4-s − 5·5-s + 6.98·6-s + 22.4·7-s + 24.6·8-s + 9·9-s + 11.6·10-s + 11·11-s + 7.72·12-s − 9.86·13-s − 52.3·14-s + 15·15-s − 36.7·16-s − 128.·17-s − 20.9·18-s + 7.04·19-s + 12.8·20-s − 67.4·21-s − 25.6·22-s + 0.654·23-s − 73.8·24-s + 25·25-s + 22.9·26-s − 27·27-s − 57.8·28-s + ⋯ |
| L(s) = 1 | − 0.823·2-s − 0.577·3-s − 0.321·4-s − 0.447·5-s + 0.475·6-s + 1.21·7-s + 1.08·8-s + 0.333·9-s + 0.368·10-s + 0.301·11-s + 0.185·12-s − 0.210·13-s − 0.998·14-s + 0.258·15-s − 0.574·16-s − 1.82·17-s − 0.274·18-s + 0.0850·19-s + 0.143·20-s − 0.700·21-s − 0.248·22-s + 0.00593·23-s − 0.628·24-s + 0.200·25-s + 0.173·26-s − 0.192·27-s − 0.390·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 + 5T \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 + 2.32T + 8T^{2} \) |
| 7 | \( 1 - 22.4T + 343T^{2} \) |
| 13 | \( 1 + 9.86T + 2.19e3T^{2} \) |
| 17 | \( 1 + 128.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 7.04T + 6.85e3T^{2} \) |
| 23 | \( 1 - 0.654T + 1.21e4T^{2} \) |
| 29 | \( 1 + 229.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 155.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 110.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 154.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 401.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 277.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 651.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 423.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 681.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 374.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 96.6T + 3.57e5T^{2} \) |
| 73 | \( 1 + 19.9T + 3.89e5T^{2} \) |
| 79 | \( 1 - 24.4T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.12e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 639.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 730.T + 9.12e5T^{2} \) |
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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
−11.38396107918053199134896155671, −11.06426070673374402137520530015, −9.784012941071971075327506411725, −8.712145943629755070877300170293, −7.901200733731107632500869835234, −6.78837948449091974826726938817, −5.05832509076758630826783281233, −4.23912785895003769931341466511, −1.66260061704225131346789765018, 0,
1.66260061704225131346789765018, 4.23912785895003769931341466511, 5.05832509076758630826783281233, 6.78837948449091974826726938817, 7.901200733731107632500869835234, 8.712145943629755070877300170293, 9.784012941071971075327506411725, 11.06426070673374402137520530015, 11.38396107918053199134896155671
