| L(s) = 1 | − 2.52·3-s − 3.46·7-s + 3.37·9-s + 11-s − 5.04·17-s − 4·19-s + 8.74·21-s + 2.52·23-s − 0.939·27-s − 2.74·29-s + 2.37·31-s − 2.52·33-s − 11.0·37-s − 2.74·41-s − 3.46·43-s − 6.63·47-s + 4.99·49-s + 12.7·51-s − 3.16·53-s + 10.0·57-s + 1.62·59-s + 10.7·61-s − 11.6·63-s + 0.644·67-s − 6.37·69-s − 7.11·71-s − 6.92·73-s + ⋯ |
| L(s) = 1 | − 1.45·3-s − 1.30·7-s + 1.12·9-s + 0.301·11-s − 1.22·17-s − 0.917·19-s + 1.90·21-s + 0.526·23-s − 0.180·27-s − 0.509·29-s + 0.426·31-s − 0.439·33-s − 1.81·37-s − 0.428·41-s − 0.528·43-s − 0.967·47-s + 0.714·49-s + 1.78·51-s − 0.435·53-s + 1.33·57-s + 0.211·59-s + 1.37·61-s − 1.47·63-s + 0.0787·67-s − 0.767·69-s − 0.844·71-s − 0.810·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3401442000\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3401442000\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 2.52T + 3T^{2} \) |
| 7 | \( 1 + 3.46T + 7T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 5.04T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 2.52T + 23T^{2} \) |
| 29 | \( 1 + 2.74T + 29T^{2} \) |
| 31 | \( 1 - 2.37T + 31T^{2} \) |
| 37 | \( 1 + 11.0T + 37T^{2} \) |
| 41 | \( 1 + 2.74T + 41T^{2} \) |
| 43 | \( 1 + 3.46T + 43T^{2} \) |
| 47 | \( 1 + 6.63T + 47T^{2} \) |
| 53 | \( 1 + 3.16T + 53T^{2} \) |
| 59 | \( 1 - 1.62T + 59T^{2} \) |
| 61 | \( 1 - 10.7T + 61T^{2} \) |
| 67 | \( 1 - 0.644T + 67T^{2} \) |
| 71 | \( 1 + 7.11T + 71T^{2} \) |
| 73 | \( 1 + 6.92T + 73T^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 - 6.63T + 83T^{2} \) |
| 89 | \( 1 + 4.37T + 89T^{2} \) |
| 97 | \( 1 + 4.10T + 97T^{2} \) |
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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
−8.560598840531948640077415397712, −7.24193496219609554133439758200, −6.57810764611931938238149906954, −6.39857162952759521260551775739, −5.48735977370804065551255828749, −4.76294002453403777017830063275, −3.93613277461613877306480102414, −3.00034785218012165053470693333, −1.76165330368244171048515843815, −0.34211179336459940434054612351,
0.34211179336459940434054612351, 1.76165330368244171048515843815, 3.00034785218012165053470693333, 3.93613277461613877306480102414, 4.76294002453403777017830063275, 5.48735977370804065551255828749, 6.39857162952759521260551775739, 6.57810764611931938238149906954, 7.24193496219609554133439758200, 8.560598840531948640077415397712
