| L(s) = 1 | + 3-s + 0.725·5-s − 3.98·7-s + 9-s + 0.244·11-s + 4.54·13-s + 0.725·15-s − 5.80·17-s − 3.98·21-s − 0.414·23-s − 4.47·25-s + 27-s − 6.43·29-s + 3.70·31-s + 0.244·33-s − 2.89·35-s + 6.12·37-s + 4.54·39-s + 9.22·41-s + 4.21·43-s + 0.725·45-s + 5.09·47-s + 8.86·49-s − 5.80·51-s + 1.12·53-s + 0.177·55-s − 1.50·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.324·5-s − 1.50·7-s + 0.333·9-s + 0.0736·11-s + 1.26·13-s + 0.187·15-s − 1.40·17-s − 0.869·21-s − 0.0864·23-s − 0.894·25-s + 0.192·27-s − 1.19·29-s + 0.666·31-s + 0.0425·33-s − 0.488·35-s + 1.00·37-s + 0.728·39-s + 1.44·41-s + 0.643·43-s + 0.108·45-s + 0.742·47-s + 1.26·49-s − 0.812·51-s + 0.154·53-s + 0.0238·55-s − 0.195·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 - 0.725T + 5T^{2} \) |
| 7 | \( 1 + 3.98T + 7T^{2} \) |
| 11 | \( 1 - 0.244T + 11T^{2} \) |
| 13 | \( 1 - 4.54T + 13T^{2} \) |
| 17 | \( 1 + 5.80T + 17T^{2} \) |
| 23 | \( 1 + 0.414T + 23T^{2} \) |
| 29 | \( 1 + 6.43T + 29T^{2} \) |
| 31 | \( 1 - 3.70T + 31T^{2} \) |
| 37 | \( 1 - 6.12T + 37T^{2} \) |
| 41 | \( 1 - 9.22T + 41T^{2} \) |
| 43 | \( 1 - 4.21T + 43T^{2} \) |
| 47 | \( 1 - 5.09T + 47T^{2} \) |
| 53 | \( 1 - 1.12T + 53T^{2} \) |
| 59 | \( 1 + 1.50T + 59T^{2} \) |
| 61 | \( 1 + 10.0T + 61T^{2} \) |
| 67 | \( 1 - 14.4T + 67T^{2} \) |
| 71 | \( 1 + 7.20T + 71T^{2} \) |
| 73 | \( 1 + 10.6T + 73T^{2} \) |
| 79 | \( 1 + 1.57T + 79T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 + 16.4T + 89T^{2} \) |
| 97 | \( 1 + 5.01T + 97T^{2} \) |
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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
−7.36289584027225111029593133106, −6.69035702975665607012505184369, −6.07214882593221255594319507445, −5.66440042605384834929205220420, −4.13027137465235306641214378408, −4.05526979755445953678921875330, −2.95605628302064226315257733251, −2.41605436783364866084588692232, −1.31178256763126377081003483962, 0,
1.31178256763126377081003483962, 2.41605436783364866084588692232, 2.95605628302064226315257733251, 4.05526979755445953678921875330, 4.13027137465235306641214378408, 5.66440042605384834929205220420, 6.07214882593221255594319507445, 6.69035702975665607012505184369, 7.36289584027225111029593133106
