| L(s) = 1 | + 2.21·2-s + 2.91·4-s − 4.37·5-s + 0.199·7-s + 2.01·8-s − 9.69·10-s + 11-s + 6.55·13-s + 0.442·14-s − 1.35·16-s − 0.167·17-s − 1.49·19-s − 12.7·20-s + 2.21·22-s − 8.33·23-s + 14.1·25-s + 14.5·26-s + 0.581·28-s + 1.38·29-s − 1.87·31-s − 7.02·32-s − 0.370·34-s − 0.873·35-s + 8.78·37-s − 3.31·38-s − 8.82·40-s − 3.86·41-s + ⋯ |
| L(s) = 1 | + 1.56·2-s + 1.45·4-s − 1.95·5-s + 0.0754·7-s + 0.713·8-s − 3.06·10-s + 0.301·11-s + 1.81·13-s + 0.118·14-s − 0.337·16-s − 0.0405·17-s − 0.343·19-s − 2.84·20-s + 0.472·22-s − 1.73·23-s + 2.82·25-s + 2.84·26-s + 0.109·28-s + 0.256·29-s − 0.335·31-s − 1.24·32-s − 0.0636·34-s − 0.147·35-s + 1.44·37-s − 0.537·38-s − 1.39·40-s − 0.603·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 - 2.21T + 2T^{2} \) |
| 5 | \( 1 + 4.37T + 5T^{2} \) |
| 7 | \( 1 - 0.199T + 7T^{2} \) |
| 13 | \( 1 - 6.55T + 13T^{2} \) |
| 17 | \( 1 + 0.167T + 17T^{2} \) |
| 19 | \( 1 + 1.49T + 19T^{2} \) |
| 23 | \( 1 + 8.33T + 23T^{2} \) |
| 29 | \( 1 - 1.38T + 29T^{2} \) |
| 31 | \( 1 + 1.87T + 31T^{2} \) |
| 37 | \( 1 - 8.78T + 37T^{2} \) |
| 41 | \( 1 + 3.86T + 41T^{2} \) |
| 43 | \( 1 + 11.0T + 43T^{2} \) |
| 47 | \( 1 + 1.72T + 47T^{2} \) |
| 53 | \( 1 + 8.30T + 53T^{2} \) |
| 59 | \( 1 - 1.27T + 59T^{2} \) |
| 67 | \( 1 - 8.70T + 67T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 + 6.82T + 73T^{2} \) |
| 79 | \( 1 + 13.5T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 - 1.60T + 89T^{2} \) |
| 97 | \( 1 - 6.07T + 97T^{2} \) |
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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
−7.71521626721078782259058297954, −6.60970195765245818187127282991, −6.41578145878224523182514085895, −5.40274468451660465847894325618, −4.54161209877255250576055680237, −3.95372183012080705626485228037, −3.65331982849824520458991395677, −2.87374782806222790344772090164, −1.49437974352894939421795490185, 0,
1.49437974352894939421795490185, 2.87374782806222790344772090164, 3.65331982849824520458991395677, 3.95372183012080705626485228037, 4.54161209877255250576055680237, 5.40274468451660465847894325618, 6.41578145878224523182514085895, 6.60970195765245818187127282991, 7.71521626721078782259058297954
