L(s) = 1 | + 1.17·2-s − 0.937·3-s − 0.613·4-s + 2.42·5-s − 1.10·6-s − 0.490·7-s − 3.07·8-s − 2.12·9-s + 2.85·10-s − 11-s + 0.575·12-s − 2.96·13-s − 0.578·14-s − 2.27·15-s − 2.39·16-s + 17-s − 2.49·18-s − 0.937·19-s − 1.48·20-s + 0.460·21-s − 1.17·22-s − 3.83·23-s + 2.88·24-s + 0.868·25-s − 3.49·26-s + 4.80·27-s + 0.301·28-s + ⋯ |
L(s) = 1 | + 0.832·2-s − 0.541·3-s − 0.306·4-s + 1.08·5-s − 0.450·6-s − 0.185·7-s − 1.08·8-s − 0.706·9-s + 0.902·10-s − 0.301·11-s + 0.166·12-s − 0.822·13-s − 0.154·14-s − 0.586·15-s − 0.599·16-s + 0.242·17-s − 0.588·18-s − 0.214·19-s − 0.332·20-s + 0.100·21-s − 0.251·22-s − 0.798·23-s + 0.589·24-s + 0.173·25-s − 0.685·26-s + 0.924·27-s + 0.0569·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.522025898\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.522025898\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 43 | \( 1 + T \) |
good | 2 | \( 1 - 1.17T + 2T^{2} \) |
| 3 | \( 1 + 0.937T + 3T^{2} \) |
| 5 | \( 1 - 2.42T + 5T^{2} \) |
| 7 | \( 1 + 0.490T + 7T^{2} \) |
| 13 | \( 1 + 2.96T + 13T^{2} \) |
| 19 | \( 1 + 0.937T + 19T^{2} \) |
| 23 | \( 1 + 3.83T + 23T^{2} \) |
| 29 | \( 1 + 1.30T + 29T^{2} \) |
| 31 | \( 1 - 2.86T + 31T^{2} \) |
| 37 | \( 1 - 2.32T + 37T^{2} \) |
| 41 | \( 1 + 2.18T + 41T^{2} \) |
| 47 | \( 1 + 8.29T + 47T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 + 2.07T + 59T^{2} \) |
| 61 | \( 1 - 3.29T + 61T^{2} \) |
| 67 | \( 1 - 13.9T + 67T^{2} \) |
| 71 | \( 1 - 9.75T + 71T^{2} \) |
| 73 | \( 1 + 1.85T + 73T^{2} \) |
| 79 | \( 1 - 13.8T + 79T^{2} \) |
| 83 | \( 1 + 2.99T + 83T^{2} \) |
| 89 | \( 1 - 1.39T + 89T^{2} \) |
| 97 | \( 1 + 7.16T + 97T^{2} \) |
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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
−7.898887054087232056456710014053, −6.66219449314454701039830041137, −6.32120615782336471641826243500, −5.44656148763070088447542040445, −5.32129058445134309995992176639, −4.48723497937681057994596356504, −3.56392159479039286991316535678, −2.73129559462895537177493458907, −2.03613483584472589989031766296, −0.52541808400228458178866384138,
0.52541808400228458178866384138, 2.03613483584472589989031766296, 2.73129559462895537177493458907, 3.56392159479039286991316535678, 4.48723497937681057994596356504, 5.32129058445134309995992176639, 5.44656148763070088447542040445, 6.32120615782336471641826243500, 6.66219449314454701039830041137, 7.898887054087232056456710014053