| L(s) = 1 | − 1.89·3-s − 5-s − 2.74·7-s + 0.574·9-s − 0.196·13-s + 1.89·15-s + 1.46·17-s − 2.68·19-s + 5.18·21-s − 1.81·23-s + 25-s + 4.58·27-s − 1.37·29-s − 5.90·31-s + 2.74·35-s + 4.57·37-s + 0.372·39-s − 1.23·41-s + 2.96·43-s − 0.574·45-s − 9.30·47-s + 0.511·49-s − 2.77·51-s + 2.69·53-s + 5.06·57-s − 14.2·59-s − 7.54·61-s + ⋯ |
| L(s) = 1 | − 1.09·3-s − 0.447·5-s − 1.03·7-s + 0.191·9-s − 0.0545·13-s + 0.488·15-s + 0.356·17-s − 0.615·19-s + 1.13·21-s − 0.378·23-s + 0.200·25-s + 0.882·27-s − 0.256·29-s − 1.06·31-s + 0.463·35-s + 0.752·37-s + 0.0595·39-s − 0.192·41-s + 0.452·43-s − 0.0856·45-s − 1.35·47-s + 0.0731·49-s − 0.388·51-s + 0.370·53-s + 0.671·57-s − 1.84·59-s − 0.965·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2834786450\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2834786450\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 1.89T + 3T^{2} \) |
| 7 | \( 1 + 2.74T + 7T^{2} \) |
| 13 | \( 1 + 0.196T + 13T^{2} \) |
| 17 | \( 1 - 1.46T + 17T^{2} \) |
| 19 | \( 1 + 2.68T + 19T^{2} \) |
| 23 | \( 1 + 1.81T + 23T^{2} \) |
| 29 | \( 1 + 1.37T + 29T^{2} \) |
| 31 | \( 1 + 5.90T + 31T^{2} \) |
| 37 | \( 1 - 4.57T + 37T^{2} \) |
| 41 | \( 1 + 1.23T + 41T^{2} \) |
| 43 | \( 1 - 2.96T + 43T^{2} \) |
| 47 | \( 1 + 9.30T + 47T^{2} \) |
| 53 | \( 1 - 2.69T + 53T^{2} \) |
| 59 | \( 1 + 14.2T + 59T^{2} \) |
| 61 | \( 1 + 7.54T + 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 - 6.87T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 - 1.87T + 79T^{2} \) |
| 83 | \( 1 + 5.79T + 83T^{2} \) |
| 89 | \( 1 - 17.3T + 89T^{2} \) |
| 97 | \( 1 + 18.3T + 97T^{2} \) |
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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
−7.55414878509966001283475917726, −6.86181995374062576212961247091, −6.18387689254745346227999808203, −5.84207474161044582667666887209, −4.97486772083407480412820664469, −4.28504933796885050629233069370, −3.44622894217803074835314973632, −2.76108442213346926883932747766, −1.51525155345407261238792498016, −0.27007992197075977548903331705,
0.27007992197075977548903331705, 1.51525155345407261238792498016, 2.76108442213346926883932747766, 3.44622894217803074835314973632, 4.28504933796885050629233069370, 4.97486772083407480412820664469, 5.84207474161044582667666887209, 6.18387689254745346227999808203, 6.86181995374062576212961247091, 7.55414878509966001283475917726
