| L(s) = 1 | − 0.963·3-s − 5.79·5-s + 15.5·7-s − 26.0·9-s + 25.2·11-s − 27.0·13-s + 5.58·15-s + 75.9·17-s − 27.9·19-s − 15.0·21-s − 23·23-s − 91.3·25-s + 51.1·27-s − 141.·29-s + 318.·31-s − 24.3·33-s − 90.4·35-s − 35.1·37-s + 26.0·39-s − 20.6·41-s − 181.·43-s + 151.·45-s + 423.·47-s − 99.8·49-s − 73.1·51-s − 167.·53-s − 146.·55-s + ⋯ |
| L(s) = 1 | − 0.185·3-s − 0.518·5-s + 0.841·7-s − 0.965·9-s + 0.693·11-s − 0.576·13-s + 0.0961·15-s + 1.08·17-s − 0.337·19-s − 0.156·21-s − 0.208·23-s − 0.731·25-s + 0.364·27-s − 0.908·29-s + 1.84·31-s − 0.128·33-s − 0.436·35-s − 0.156·37-s + 0.106·39-s − 0.0786·41-s − 0.643·43-s + 0.500·45-s + 1.31·47-s − 0.291·49-s − 0.200·51-s − 0.434·53-s − 0.359·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 23 | \( 1 + 23T \) |
| good | 3 | \( 1 + 0.963T + 27T^{2} \) |
| 5 | \( 1 + 5.79T + 125T^{2} \) |
| 7 | \( 1 - 15.5T + 343T^{2} \) |
| 11 | \( 1 - 25.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 27.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 75.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 27.9T + 6.85e3T^{2} \) |
| 29 | \( 1 + 141.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 318.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 35.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + 20.6T + 6.89e4T^{2} \) |
| 43 | \( 1 + 181.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 423.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 167.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 72.6T + 2.05e5T^{2} \) |
| 61 | \( 1 - 744.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 538.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 737.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 4.93T + 3.89e5T^{2} \) |
| 79 | \( 1 - 216.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.47e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.09e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 633.T + 9.12e5T^{2} \) |
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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
−8.543778684343953999427425212057, −8.025483896111666383238968768789, −7.24874463105913302325710790459, −6.18099746135005925089254630881, −5.41738309295523232382071590759, −4.52409747908973660481649656853, −3.61467878951172875445477524100, −2.50877504333493819021099835409, −1.26383031288071616136291189976, 0,
1.26383031288071616136291189976, 2.50877504333493819021099835409, 3.61467878951172875445477524100, 4.52409747908973660481649656853, 5.41738309295523232382071590759, 6.18099746135005925089254630881, 7.24874463105913302325710790459, 8.025483896111666383238968768789, 8.543778684343953999427425212057
