| L(s) = 1 | + 1.87·2-s + 1.53·4-s − 0.879·5-s − 2.87·7-s − 0.879·8-s − 1.65·10-s + 1.83·11-s + 2.75·13-s − 5.41·14-s − 4.71·16-s + 7.10·17-s − 1.34·20-s + 3.45·22-s − 6.59·23-s − 4.22·25-s + 5.18·26-s − 4.41·28-s − 3.12·29-s − 7.65·31-s − 7.10·32-s + 13.3·34-s + 2.53·35-s − 2.83·37-s + 0.773·40-s + 3.98·41-s − 11.4·43-s + 2.81·44-s + ⋯ |
| L(s) = 1 | + 1.32·2-s + 0.766·4-s − 0.393·5-s − 1.08·7-s − 0.310·8-s − 0.522·10-s + 0.554·11-s + 0.765·13-s − 1.44·14-s − 1.17·16-s + 1.72·17-s − 0.301·20-s + 0.736·22-s − 1.37·23-s − 0.845·25-s + 1.01·26-s − 0.833·28-s − 0.580·29-s − 1.37·31-s − 1.25·32-s + 2.29·34-s + 0.428·35-s − 0.466·37-s + 0.122·40-s + 0.622·41-s − 1.74·43-s + 0.424·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 1.87T + 2T^{2} \) |
| 5 | \( 1 + 0.879T + 5T^{2} \) |
| 7 | \( 1 + 2.87T + 7T^{2} \) |
| 11 | \( 1 - 1.83T + 11T^{2} \) |
| 13 | \( 1 - 2.75T + 13T^{2} \) |
| 17 | \( 1 - 7.10T + 17T^{2} \) |
| 23 | \( 1 + 6.59T + 23T^{2} \) |
| 29 | \( 1 + 3.12T + 29T^{2} \) |
| 31 | \( 1 + 7.65T + 31T^{2} \) |
| 37 | \( 1 + 2.83T + 37T^{2} \) |
| 41 | \( 1 - 3.98T + 41T^{2} \) |
| 43 | \( 1 + 11.4T + 43T^{2} \) |
| 47 | \( 1 + 2.20T + 47T^{2} \) |
| 53 | \( 1 - 2.70T + 53T^{2} \) |
| 59 | \( 1 + 8.41T + 59T^{2} \) |
| 61 | \( 1 - 0.615T + 61T^{2} \) |
| 67 | \( 1 - 3.67T + 67T^{2} \) |
| 71 | \( 1 + 7.45T + 71T^{2} \) |
| 73 | \( 1 + 10.0T + 73T^{2} \) |
| 79 | \( 1 + 1.61T + 79T^{2} \) |
| 83 | \( 1 + 0.985T + 83T^{2} \) |
| 89 | \( 1 - 17.0T + 89T^{2} \) |
| 97 | \( 1 - 5.90T + 97T^{2} \) |
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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
−8.144567909702436092444751057699, −7.35638243871927594212006630445, −6.44017027180021526906221182980, −5.91098513976299248192409923333, −5.31962552122786012169430928551, −4.11216525929314633633558445792, −3.61668295902895967167642784199, −3.14267069759827921303083271529, −1.72983484977911585945332139552, 0,
1.72983484977911585945332139552, 3.14267069759827921303083271529, 3.61668295902895967167642784199, 4.11216525929314633633558445792, 5.31962552122786012169430928551, 5.91098513976299248192409923333, 6.44017027180021526906221182980, 7.35638243871927594212006630445, 8.144567909702436092444751057699
