| L(s) = 1 | − 0.500·2-s − 1.74·4-s + 0.0237·7-s + 1.87·8-s − 3.58·11-s + 3.77·13-s − 0.0119·14-s + 2.55·16-s + 3.62·17-s − 2.43·19-s + 1.79·22-s − 1.71·23-s − 1.89·26-s − 0.0416·28-s − 3.85·29-s − 6.00·31-s − 5.03·32-s − 1.81·34-s − 0.369·37-s + 1.21·38-s + 7.80·41-s + 0.174·43-s + 6.26·44-s + 0.859·46-s − 7.81·47-s − 6.99·49-s − 6.60·52-s + ⋯ |
| L(s) = 1 | − 0.354·2-s − 0.874·4-s + 0.00899·7-s + 0.664·8-s − 1.08·11-s + 1.04·13-s − 0.00318·14-s + 0.639·16-s + 0.878·17-s − 0.557·19-s + 0.382·22-s − 0.357·23-s − 0.370·26-s − 0.00786·28-s − 0.716·29-s − 1.07·31-s − 0.890·32-s − 0.311·34-s − 0.0607·37-s + 0.197·38-s + 1.21·41-s + 0.0266·43-s + 0.944·44-s + 0.126·46-s − 1.13·47-s − 0.999·49-s − 0.915·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 0.500T + 2T^{2} \) |
| 7 | \( 1 - 0.0237T + 7T^{2} \) |
| 11 | \( 1 + 3.58T + 11T^{2} \) |
| 13 | \( 1 - 3.77T + 13T^{2} \) |
| 17 | \( 1 - 3.62T + 17T^{2} \) |
| 19 | \( 1 + 2.43T + 19T^{2} \) |
| 23 | \( 1 + 1.71T + 23T^{2} \) |
| 29 | \( 1 + 3.85T + 29T^{2} \) |
| 31 | \( 1 + 6.00T + 31T^{2} \) |
| 37 | \( 1 + 0.369T + 37T^{2} \) |
| 41 | \( 1 - 7.80T + 41T^{2} \) |
| 43 | \( 1 - 0.174T + 43T^{2} \) |
| 47 | \( 1 + 7.81T + 47T^{2} \) |
| 53 | \( 1 - 8.97T + 53T^{2} \) |
| 59 | \( 1 - 4.45T + 59T^{2} \) |
| 61 | \( 1 - 9.21T + 61T^{2} \) |
| 67 | \( 1 + 4.47T + 67T^{2} \) |
| 71 | \( 1 - 9.69T + 71T^{2} \) |
| 73 | \( 1 + 3.95T + 73T^{2} \) |
| 79 | \( 1 + 9.68T + 79T^{2} \) |
| 83 | \( 1 - 8.95T + 83T^{2} \) |
| 89 | \( 1 - 17.0T + 89T^{2} \) |
| 97 | \( 1 - 2.76T + 97T^{2} \) |
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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
−7.910571821960216419441349699134, −7.35257651582760174061446908629, −6.24932900510080543850764867620, −5.53090557991723215611303750364, −4.98509193723313395539155555563, −3.96807121190676809235900730714, −3.45520849804221101639911822872, −2.23739490880196254982875977419, −1.15289605339462428097022937967, 0,
1.15289605339462428097022937967, 2.23739490880196254982875977419, 3.45520849804221101639911822872, 3.96807121190676809235900730714, 4.98509193723313395539155555563, 5.53090557991723215611303750364, 6.24932900510080543850764867620, 7.35257651582760174061446908629, 7.910571821960216419441349699134
