| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s + 2.55·7-s − 8-s + 9-s − 10-s + 2.24·11-s − 12-s − 2.55·14-s − 15-s + 16-s + 2.02·17-s − 18-s − 1.33·19-s + 20-s − 2.55·21-s − 2.24·22-s + 5.58·23-s + 24-s + 25-s − 27-s + 2.55·28-s + 3.26·29-s + 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s + 0.965·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s + 0.677·11-s − 0.288·12-s − 0.682·14-s − 0.258·15-s + 0.250·16-s + 0.490·17-s − 0.235·18-s − 0.306·19-s + 0.223·20-s − 0.557·21-s − 0.479·22-s + 1.16·23-s + 0.204·24-s + 0.200·25-s − 0.192·27-s + 0.482·28-s + 0.606·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.628592392\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.628592392\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 - 2.55T + 7T^{2} \) |
| 11 | \( 1 - 2.24T + 11T^{2} \) |
| 17 | \( 1 - 2.02T + 17T^{2} \) |
| 19 | \( 1 + 1.33T + 19T^{2} \) |
| 23 | \( 1 - 5.58T + 23T^{2} \) |
| 29 | \( 1 - 3.26T + 29T^{2} \) |
| 31 | \( 1 + 1.35T + 31T^{2} \) |
| 37 | \( 1 - 9.64T + 37T^{2} \) |
| 41 | \( 1 - 1.06T + 41T^{2} \) |
| 43 | \( 1 - 0.137T + 43T^{2} \) |
| 47 | \( 1 + 12.7T + 47T^{2} \) |
| 53 | \( 1 + 3.03T + 53T^{2} \) |
| 59 | \( 1 - 11.9T + 59T^{2} \) |
| 61 | \( 1 + 0.972T + 61T^{2} \) |
| 67 | \( 1 - 8.51T + 67T^{2} \) |
| 71 | \( 1 - 2.69T + 71T^{2} \) |
| 73 | \( 1 - 1.48T + 73T^{2} \) |
| 79 | \( 1 + 7.77T + 79T^{2} \) |
| 83 | \( 1 - 17.7T + 83T^{2} \) |
| 89 | \( 1 - 15.4T + 89T^{2} \) |
| 97 | \( 1 + 7.14T + 97T^{2} \) |
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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
−8.189927110023359231292118961198, −7.64143007298733193025077096260, −6.69419979985273590605932872251, −6.29582689258910157218267742706, −5.29081711185365225161889679601, −4.77747145864549295479881626556, −3.73125113921214843641479629738, −2.59426246880366383518821996085, −1.59279655512156638292963990338, −0.871005823326946464219427606020,
0.871005823326946464219427606020, 1.59279655512156638292963990338, 2.59426246880366383518821996085, 3.73125113921214843641479629738, 4.77747145864549295479881626556, 5.29081711185365225161889679601, 6.29582689258910157218267742706, 6.69419979985273590605932872251, 7.64143007298733193025077096260, 8.189927110023359231292118961198
