| L(s) = 1 | + 1.95·2-s + 0.296·3-s + 1.82·4-s − 5-s + 0.580·6-s − 3.56·7-s − 0.340·8-s − 2.91·9-s − 1.95·10-s + 5.56·11-s + 0.541·12-s + 5.26·13-s − 6.96·14-s − 0.296·15-s − 4.31·16-s + 1.40·17-s − 5.69·18-s + 19-s − 1.82·20-s − 1.05·21-s + 10.8·22-s − 6.96·23-s − 0.101·24-s + 25-s + 10.3·26-s − 1.75·27-s − 6.50·28-s + ⋯ |
| L(s) = 1 | + 1.38·2-s + 0.171·3-s + 0.912·4-s − 0.447·5-s + 0.237·6-s − 1.34·7-s − 0.120·8-s − 0.970·9-s − 0.618·10-s + 1.67·11-s + 0.156·12-s + 1.46·13-s − 1.86·14-s − 0.0766·15-s − 1.07·16-s + 0.341·17-s − 1.34·18-s + 0.229·19-s − 0.408·20-s − 0.230·21-s + 2.31·22-s − 1.45·23-s − 0.0206·24-s + 0.200·25-s + 2.02·26-s − 0.337·27-s − 1.22·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.661186439\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.661186439\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 1.95T + 2T^{2} \) |
| 3 | \( 1 - 0.296T + 3T^{2} \) |
| 7 | \( 1 + 3.56T + 7T^{2} \) |
| 11 | \( 1 - 5.56T + 11T^{2} \) |
| 13 | \( 1 - 5.26T + 13T^{2} \) |
| 17 | \( 1 - 1.40T + 17T^{2} \) |
| 23 | \( 1 + 6.96T + 23T^{2} \) |
| 29 | \( 1 - 1.40T + 29T^{2} \) |
| 31 | \( 1 - 1.75T + 31T^{2} \) |
| 37 | \( 1 - 3.61T + 37T^{2} \) |
| 41 | \( 1 - 4.34T + 41T^{2} \) |
| 43 | \( 1 + 3.56T + 43T^{2} \) |
| 47 | \( 1 + 8.26T + 47T^{2} \) |
| 53 | \( 1 + 7.61T + 53T^{2} \) |
| 59 | \( 1 - 9.47T + 59T^{2} \) |
| 61 | \( 1 - 9.21T + 61T^{2} \) |
| 67 | \( 1 + 4.76T + 67T^{2} \) |
| 71 | \( 1 + 14.0T + 71T^{2} \) |
| 73 | \( 1 - 6.59T + 73T^{2} \) |
| 79 | \( 1 - 5.47T + 79T^{2} \) |
| 83 | \( 1 + 4.15T + 83T^{2} \) |
| 89 | \( 1 + 9.23T + 89T^{2} \) |
| 97 | \( 1 - 11.5T + 97T^{2} \) |
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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
−13.98605755115621292932543297015, −13.09967343822855024390102817561, −12.02278327921214805837664331646, −11.37912829599247876963620495499, −9.585017217411851428859722646500, −8.507476021483457623575999730589, −6.52380520241516464012175824340, −5.95286631379244483610738474662, −3.99159469342886300487367493015, −3.26205793566013500296942738648,
3.26205793566013500296942738648, 3.99159469342886300487367493015, 5.95286631379244483610738474662, 6.52380520241516464012175824340, 8.507476021483457623575999730589, 9.585017217411851428859722646500, 11.37912829599247876963620495499, 12.02278327921214805837664331646, 13.09967343822855024390102817561, 13.98605755115621292932543297015
