| L(s) = 1 | − 60.4·2-s − 54.5·3-s + 1.60e3·4-s − 233.·5-s + 3.29e3·6-s + 8.14e4·7-s + 2.65e4·8-s − 1.74e5·9-s + 1.41e4·10-s − 8.42e5·11-s − 8.77e4·12-s + 3.71e5·13-s − 4.92e6·14-s + 1.27e4·15-s − 4.90e6·16-s − 4.26e5·17-s + 1.05e7·18-s − 5.38e6·19-s − 3.75e5·20-s − 4.44e6·21-s + 5.09e7·22-s + 4.61e6·23-s − 1.44e6·24-s − 4.87e7·25-s − 2.24e7·26-s + 1.91e7·27-s + 1.31e8·28-s + ⋯ |
| L(s) = 1 | − 1.33·2-s − 0.129·3-s + 0.785·4-s − 0.0334·5-s + 0.173·6-s + 1.83·7-s + 0.286·8-s − 0.983·9-s + 0.0446·10-s − 1.57·11-s − 0.101·12-s + 0.277·13-s − 2.44·14-s + 0.00433·15-s − 1.16·16-s − 0.0729·17-s + 1.31·18-s − 0.499·19-s − 0.0262·20-s − 0.237·21-s + 2.10·22-s + 0.149·23-s − 0.0371·24-s − 0.998·25-s − 0.370·26-s + 0.256·27-s + 1.43·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 - 3.71e5T \) |
| good | 2 | \( 1 + 60.4T + 2.04e3T^{2} \) |
| 3 | \( 1 + 54.5T + 1.77e5T^{2} \) |
| 5 | \( 1 + 233.T + 4.88e7T^{2} \) |
| 7 | \( 1 - 8.14e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 8.42e5T + 2.85e11T^{2} \) |
| 17 | \( 1 + 4.26e5T + 3.42e13T^{2} \) |
| 19 | \( 1 + 5.38e6T + 1.16e14T^{2} \) |
| 23 | \( 1 - 4.61e6T + 9.52e14T^{2} \) |
| 29 | \( 1 + 1.43e8T + 1.22e16T^{2} \) |
| 31 | \( 1 - 1.11e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 6.33e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 1.31e9T + 5.50e17T^{2} \) |
| 43 | \( 1 - 4.05e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + 1.42e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 1.08e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + 6.04e9T + 3.01e19T^{2} \) |
| 61 | \( 1 + 2.92e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 8.45e9T + 1.22e20T^{2} \) |
| 71 | \( 1 - 5.43e8T + 2.31e20T^{2} \) |
| 73 | \( 1 - 4.54e9T + 3.13e20T^{2} \) |
| 79 | \( 1 - 1.08e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 2.94e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 1.13e8T + 2.77e21T^{2} \) |
| 97 | \( 1 - 1.23e11T + 7.15e21T^{2} \) |
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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
−17.02926814692525170445931299710, −15.35298871180518818734152095001, −13.76603645503437829109474944215, −11.44605060115710624950503296833, −10.52780100888208740487924857803, −8.562249301404546238285599379070, −7.79194198074640851283224250125, −5.13719408160689672618511782328, −1.91572068335219604699466681664, 0,
1.91572068335219604699466681664, 5.13719408160689672618511782328, 7.79194198074640851283224250125, 8.562249301404546238285599379070, 10.52780100888208740487924857803, 11.44605060115710624950503296833, 13.76603645503437829109474944215, 15.35298871180518818734152095001, 17.02926814692525170445931299710
