| L(s) = 1 | − 2.92·5-s − 7-s + 2.82·11-s + 6.13·13-s − 2.92·17-s + 19-s − 2.82·23-s + 3.54·25-s − 3.90·29-s − 6.96·31-s + 2.92·35-s − 3.65·37-s − 8.42·41-s + 1.30·43-s + 5.59·47-s + 49-s + 10.0·53-s − 8.26·55-s + 6.60·59-s + 12.7·61-s − 17.9·65-s + 8.48·67-s − 10.3·71-s − 1.81·73-s − 2.82·77-s − 2.06·79-s − 6.79·83-s + ⋯ |
| L(s) = 1 | − 1.30·5-s − 0.377·7-s + 0.852·11-s + 1.70·13-s − 0.708·17-s + 0.229·19-s − 0.589·23-s + 0.708·25-s − 0.725·29-s − 1.25·31-s + 0.493·35-s − 0.601·37-s − 1.31·41-s + 0.198·43-s + 0.816·47-s + 0.142·49-s + 1.37·53-s − 1.11·55-s + 0.860·59-s + 1.63·61-s − 2.22·65-s + 1.03·67-s − 1.22·71-s − 0.212·73-s − 0.322·77-s − 0.232·79-s − 0.746·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 2.92T + 5T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 - 6.13T + 13T^{2} \) |
| 17 | \( 1 + 2.92T + 17T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 + 3.90T + 29T^{2} \) |
| 31 | \( 1 + 6.96T + 31T^{2} \) |
| 37 | \( 1 + 3.65T + 37T^{2} \) |
| 41 | \( 1 + 8.42T + 41T^{2} \) |
| 43 | \( 1 - 1.30T + 43T^{2} \) |
| 47 | \( 1 - 5.59T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 - 6.60T + 59T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 - 8.48T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 + 1.81T + 73T^{2} \) |
| 79 | \( 1 + 2.06T + 79T^{2} \) |
| 83 | \( 1 + 6.79T + 83T^{2} \) |
| 89 | \( 1 - 8.11T + 89T^{2} \) |
| 97 | \( 1 + 6.67T + 97T^{2} \) |
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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
−7.18613268820446742477737861455, −6.87051170567940401260615714752, −6.01266229026641322878820013531, −5.37472978649833913411920139629, −4.21495086387384534317996969503, −3.78875169403962286595979090085, −3.43140160804472418412712420629, −2.12442599031299985030138064756, −1.10248022934404392166734604299, 0,
1.10248022934404392166734604299, 2.12442599031299985030138064756, 3.43140160804472418412712420629, 3.78875169403962286595979090085, 4.21495086387384534317996969503, 5.37472978649833913411920139629, 6.01266229026641322878820013531, 6.87051170567940401260615714752, 7.18613268820446742477737861455
