| L(s) = 1 | − 1.62·2-s − 9.62·3-s − 29.3·4-s − 25·5-s + 15.6·6-s + 193.·7-s + 99.7·8-s − 150.·9-s + 40.6·10-s + 282.·12-s − 320.·13-s − 313.·14-s + 240.·15-s + 777.·16-s − 710.·17-s + 244.·18-s − 2.30e3·19-s + 733.·20-s − 1.85e3·21-s + 552.·23-s − 959.·24-s + 625·25-s + 520.·26-s + 3.78e3·27-s − 5.66e3·28-s + 3.93e3·29-s − 391.·30-s + ⋯ |
| L(s) = 1 | − 0.287·2-s − 0.617·3-s − 0.917·4-s − 0.447·5-s + 0.177·6-s + 1.48·7-s + 0.551·8-s − 0.618·9-s + 0.128·10-s + 0.566·12-s − 0.525·13-s − 0.427·14-s + 0.276·15-s + 0.759·16-s − 0.595·17-s + 0.177·18-s − 1.46·19-s + 0.410·20-s − 0.919·21-s + 0.217·23-s − 0.340·24-s + 0.200·25-s + 0.151·26-s + 0.999·27-s − 1.36·28-s + 0.867·29-s − 0.0793·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + 25T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.62T + 32T^{2} \) |
| 3 | \( 1 + 9.62T + 243T^{2} \) |
| 7 | \( 1 - 193.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 320.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 710.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.30e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 552.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.93e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.64e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 3.76e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 2.34e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.65e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.61e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.14e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.99e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.59e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.19e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 7.16e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.84e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.16e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.74e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.71e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 6.48e4T + 8.58e9T^{2} \) |
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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
−9.279178017828446072524956191491, −8.400054251155561347250512980372, −8.043013426502179142798135528529, −6.79182691240033359733422151395, −5.54903201437152457315064641606, −4.74250396094015109223606347612, −4.15990547269945700192686569956, −2.41974369421096816004713520078, −1.00151569567688055602869643507, 0,
1.00151569567688055602869643507, 2.41974369421096816004713520078, 4.15990547269945700192686569956, 4.74250396094015109223606347612, 5.54903201437152457315064641606, 6.79182691240033359733422151395, 8.043013426502179142798135528529, 8.400054251155561347250512980372, 9.279178017828446072524956191491
