| L(s) = 1 | − 1.48·5-s − 3.15·11-s + 6.76·13-s − 2·17-s − 1.03·19-s − 4.24·23-s − 2.79·25-s − 29-s − 1.87·31-s − 0.969·37-s + 7.52·41-s + 1.09·43-s + 9.34·47-s − 7·49-s − 5.73·53-s + 4.68·55-s − 8.24·59-s − 10.4·61-s − 10.0·65-s − 4.49·67-s − 10.1·71-s + 11.5·73-s − 13.0·79-s − 16.2·83-s + 2.96·85-s − 13.5·89-s + 1.52·95-s + ⋯ |
| L(s) = 1 | − 0.664·5-s − 0.951·11-s + 1.87·13-s − 0.485·17-s − 0.236·19-s − 0.886·23-s − 0.559·25-s − 0.185·29-s − 0.336·31-s − 0.159·37-s + 1.17·41-s + 0.166·43-s + 1.36·47-s − 49-s − 0.787·53-s + 0.631·55-s − 1.07·59-s − 1.34·61-s − 1.24·65-s − 0.549·67-s − 1.20·71-s + 1.34·73-s − 1.47·79-s − 1.78·83-s + 0.322·85-s − 1.44·89-s + 0.156·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2088 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 + 1.48T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 3.15T + 11T^{2} \) |
| 13 | \( 1 - 6.76T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 1.03T + 19T^{2} \) |
| 23 | \( 1 + 4.24T + 23T^{2} \) |
| 31 | \( 1 + 1.87T + 31T^{2} \) |
| 37 | \( 1 + 0.969T + 37T^{2} \) |
| 41 | \( 1 - 7.52T + 41T^{2} \) |
| 43 | \( 1 - 1.09T + 43T^{2} \) |
| 47 | \( 1 - 9.34T + 47T^{2} \) |
| 53 | \( 1 + 5.73T + 53T^{2} \) |
| 59 | \( 1 + 8.24T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 + 4.49T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 + 13.0T + 79T^{2} \) |
| 83 | \( 1 + 16.2T + 83T^{2} \) |
| 89 | \( 1 + 13.5T + 89T^{2} \) |
| 97 | \( 1 + 4.96T + 97T^{2} \) |
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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
−8.597575858104773086151543299516, −7.986931709816135725205925670570, −7.34229928945729248462755651243, −6.19703319688135327179743465892, −5.72479697020013955130265098143, −4.46186212320372177533266659220, −3.84122964550264485555890387985, −2.85085532493849969305992105017, −1.56846702150384281821622001011, 0,
1.56846702150384281821622001011, 2.85085532493849969305992105017, 3.84122964550264485555890387985, 4.46186212320372177533266659220, 5.72479697020013955130265098143, 6.19703319688135327179743465892, 7.34229928945729248462755651243, 7.986931709816135725205925670570, 8.597575858104773086151543299516
