| L(s) = 1 | + 1.02e3·2-s − 1.20e5·3-s − 1.03e6·4-s − 1.23e8·6-s − 2.19e8·7-s − 3.22e9·8-s + 3.96e9·9-s + 1.30e11·11-s + 1.24e11·12-s + 4.83e11·13-s − 2.26e11·14-s − 1.14e12·16-s + 2.95e12·17-s + 4.07e12·18-s + 3.58e12·19-s + 2.63e13·21-s + 1.34e14·22-s − 2.51e14·23-s + 3.87e14·24-s + 4.97e14·26-s + 7.80e14·27-s + 2.28e14·28-s − 1.57e15·29-s + 7.67e15·31-s + 5.59e15·32-s − 1.56e16·33-s + 3.03e15·34-s + ⋯ |
| L(s) = 1 | + 0.710·2-s − 1.17·3-s − 0.495·4-s − 0.834·6-s − 0.294·7-s − 1.06·8-s + 0.378·9-s + 1.51·11-s + 0.581·12-s + 0.971·13-s − 0.208·14-s − 0.259·16-s + 0.354·17-s + 0.269·18-s + 0.134·19-s + 0.345·21-s + 1.07·22-s − 1.26·23-s + 1.24·24-s + 0.690·26-s + 0.729·27-s + 0.145·28-s − 0.697·29-s + 1.68·31-s + 0.877·32-s − 1.78·33-s + 0.252·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 - 1.02e3T + 2.09e6T^{2} \) |
| 3 | \( 1 + 1.20e5T + 1.04e10T^{2} \) |
| 7 | \( 1 + 2.19e8T + 5.58e17T^{2} \) |
| 11 | \( 1 - 1.30e11T + 7.40e21T^{2} \) |
| 13 | \( 1 - 4.83e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 2.95e12T + 6.90e25T^{2} \) |
| 19 | \( 1 - 3.58e12T + 7.14e26T^{2} \) |
| 23 | \( 1 + 2.51e14T + 3.94e28T^{2} \) |
| 29 | \( 1 + 1.57e15T + 5.13e30T^{2} \) |
| 31 | \( 1 - 7.67e15T + 2.08e31T^{2} \) |
| 37 | \( 1 + 3.40e15T + 8.55e32T^{2} \) |
| 41 | \( 1 + 1.11e17T + 7.38e33T^{2} \) |
| 43 | \( 1 + 6.79e15T + 2.00e34T^{2} \) |
| 47 | \( 1 - 2.04e17T + 1.30e35T^{2} \) |
| 53 | \( 1 + 1.62e18T + 1.62e36T^{2} \) |
| 59 | \( 1 - 1.65e18T + 1.54e37T^{2} \) |
| 61 | \( 1 - 8.99e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 2.86e19T + 2.22e38T^{2} \) |
| 71 | \( 1 + 3.67e19T + 7.52e38T^{2} \) |
| 73 | \( 1 + 1.46e19T + 1.34e39T^{2} \) |
| 79 | \( 1 - 4.09e19T + 7.08e39T^{2} \) |
| 83 | \( 1 + 2.10e20T + 1.99e40T^{2} \) |
| 89 | \( 1 - 3.56e20T + 8.65e40T^{2} \) |
| 97 | \( 1 + 6.40e20T + 5.27e41T^{2} \) |
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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
−12.16924208284550867050460149614, −11.53974858597068473433477581530, −9.939293837572598470001005892393, −8.605780955641735706755173053110, −6.46347943286639896313159778440, −5.81934760963163777119411545085, −4.47986527396090878252631691707, −3.43615495366534497330709638379, −1.20582680398093817706553763958, 0,
1.20582680398093817706553763958, 3.43615495366534497330709638379, 4.47986527396090878252631691707, 5.81934760963163777119411545085, 6.46347943286639896313159778440, 8.605780955641735706755173053110, 9.939293837572598470001005892393, 11.53974858597068473433477581530, 12.16924208284550867050460149614
