| L(s) = 1 | + 2-s + 0.970·3-s + 4-s + 0.208·5-s + 0.970·6-s − 0.143·7-s + 8-s − 2.05·9-s + 0.208·10-s + 11-s + 0.970·12-s − 1.77·13-s − 0.143·14-s + 0.202·15-s + 16-s − 5.73·17-s − 2.05·18-s − 2.88·19-s + 0.208·20-s − 0.138·21-s + 22-s + 1.00·23-s + 0.970·24-s − 4.95·25-s − 1.77·26-s − 4.90·27-s − 0.143·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.560·3-s + 0.5·4-s + 0.0932·5-s + 0.396·6-s − 0.0540·7-s + 0.353·8-s − 0.685·9-s + 0.0659·10-s + 0.301·11-s + 0.280·12-s − 0.491·13-s − 0.0382·14-s + 0.0522·15-s + 0.250·16-s − 1.39·17-s − 0.485·18-s − 0.662·19-s + 0.0466·20-s − 0.0303·21-s + 0.213·22-s + 0.208·23-s + 0.198·24-s − 0.991·25-s − 0.347·26-s − 0.944·27-s − 0.0270·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 197 | \( 1 + T \) |
| good | 3 | \( 1 - 0.970T + 3T^{2} \) |
| 5 | \( 1 - 0.208T + 5T^{2} \) |
| 7 | \( 1 + 0.143T + 7T^{2} \) |
| 13 | \( 1 + 1.77T + 13T^{2} \) |
| 17 | \( 1 + 5.73T + 17T^{2} \) |
| 19 | \( 1 + 2.88T + 19T^{2} \) |
| 23 | \( 1 - 1.00T + 23T^{2} \) |
| 29 | \( 1 + 5.11T + 29T^{2} \) |
| 31 | \( 1 + 3.84T + 31T^{2} \) |
| 37 | \( 1 + 3.80T + 37T^{2} \) |
| 41 | \( 1 + 12.4T + 41T^{2} \) |
| 43 | \( 1 - 5.49T + 43T^{2} \) |
| 47 | \( 1 - 3.77T + 47T^{2} \) |
| 53 | \( 1 - 4.68T + 53T^{2} \) |
| 59 | \( 1 - 8.09T + 59T^{2} \) |
| 61 | \( 1 - 9.60T + 61T^{2} \) |
| 67 | \( 1 + 7.28T + 67T^{2} \) |
| 71 | \( 1 - 5.65T + 71T^{2} \) |
| 73 | \( 1 - 7.45T + 73T^{2} \) |
| 79 | \( 1 - 12.0T + 79T^{2} \) |
| 83 | \( 1 + 0.278T + 83T^{2} \) |
| 89 | \( 1 + 15.6T + 89T^{2} \) |
| 97 | \( 1 + 10.5T + 97T^{2} \) |
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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
−8.043686977720247344821266412331, −7.13564135035451468060238519591, −6.57059611149700023382762157967, −5.70255693277503618218193379877, −5.05382887980250435843696037502, −4.05627187446891288118443671717, −3.50804750641925684323369572751, −2.41720553406537893157437068713, −1.92074364336602407451782480465, 0,
1.92074364336602407451782480465, 2.41720553406537893157437068713, 3.50804750641925684323369572751, 4.05627187446891288118443671717, 5.05382887980250435843696037502, 5.70255693277503618218193379877, 6.57059611149700023382762157967, 7.13564135035451468060238519591, 8.043686977720247344821266412331
