| L(s) = 1 | − 2.65·2-s − 2.08·3-s + 5.05·4-s − 0.388·5-s + 5.54·6-s − 5.15·7-s − 8.11·8-s + 1.36·9-s + 1.03·10-s + 3.34·11-s − 10.5·12-s + 5.68·13-s + 13.7·14-s + 0.811·15-s + 11.4·16-s − 6.13·17-s − 3.62·18-s + 2.07·19-s − 1.96·20-s + 10.7·21-s − 8.87·22-s − 3.39·23-s + 16.9·24-s − 4.84·25-s − 15.1·26-s + 3.41·27-s − 26.0·28-s + ⋯ |
| L(s) = 1 | − 1.87·2-s − 1.20·3-s + 2.52·4-s − 0.173·5-s + 2.26·6-s − 1.94·7-s − 2.86·8-s + 0.455·9-s + 0.326·10-s + 1.00·11-s − 3.04·12-s + 1.57·13-s + 3.66·14-s + 0.209·15-s + 2.85·16-s − 1.48·17-s − 0.854·18-s + 0.475·19-s − 0.438·20-s + 2.35·21-s − 1.89·22-s − 0.707·23-s + 3.45·24-s − 0.969·25-s − 2.96·26-s + 0.657·27-s − 4.92·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 37 | \( 1 + T \) |
| 109 | \( 1 + T \) |
| good | 2 | \( 1 + 2.65T + 2T^{2} \) |
| 3 | \( 1 + 2.08T + 3T^{2} \) |
| 5 | \( 1 + 0.388T + 5T^{2} \) |
| 7 | \( 1 + 5.15T + 7T^{2} \) |
| 11 | \( 1 - 3.34T + 11T^{2} \) |
| 13 | \( 1 - 5.68T + 13T^{2} \) |
| 17 | \( 1 + 6.13T + 17T^{2} \) |
| 19 | \( 1 - 2.07T + 19T^{2} \) |
| 23 | \( 1 + 3.39T + 23T^{2} \) |
| 29 | \( 1 - 1.48T + 29T^{2} \) |
| 31 | \( 1 + 7.78T + 31T^{2} \) |
| 41 | \( 1 - 4.76T + 41T^{2} \) |
| 43 | \( 1 + 0.146T + 43T^{2} \) |
| 47 | \( 1 + 5.10T + 47T^{2} \) |
| 53 | \( 1 - 1.00T + 53T^{2} \) |
| 59 | \( 1 + 1.03T + 59T^{2} \) |
| 61 | \( 1 + 3.25T + 61T^{2} \) |
| 67 | \( 1 - 11.9T + 67T^{2} \) |
| 71 | \( 1 + 3.61T + 71T^{2} \) |
| 73 | \( 1 - 12.4T + 73T^{2} \) |
| 79 | \( 1 - 4.12T + 79T^{2} \) |
| 83 | \( 1 + 15.9T + 83T^{2} \) |
| 89 | \( 1 - 9.37T + 89T^{2} \) |
| 97 | \( 1 - 0.428T + 97T^{2} \) |
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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
−8.296144975248662672470500804890, −7.19644867687925843108497300871, −6.60787198873947404315641540168, −6.23446519345382747434187537480, −5.75317922419962551935856606575, −3.99708759663790024680630200872, −3.27078955677794828898480891877, −2.00372553096685757321426569876, −0.818186439174244530158557613567, 0,
0.818186439174244530158557613567, 2.00372553096685757321426569876, 3.27078955677794828898480891877, 3.99708759663790024680630200872, 5.75317922419962551935856606575, 6.23446519345382747434187537480, 6.60787198873947404315641540168, 7.19644867687925843108497300871, 8.296144975248662672470500804890
