L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s + 3.64·7-s − 8-s + 9-s − 10-s − 1.66·11-s + 12-s − 3.64·14-s + 15-s + 16-s + 4·17-s − 18-s + 6.31·19-s + 20-s + 3.64·21-s + 1.66·22-s + 1.24·23-s − 24-s + 25-s + 27-s + 3.64·28-s + 10.0·29-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 + 1.37·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s − 0.502·11-s + 0.288·12-s − 0.974·14-s + 0.258·15-s + 0.250·16-s + 0.970·17-s − 0.235·18-s + 1.44·19-s + 0.223·20-s + 0.795·21-s + 0.355·22-s + 0.259·23-s − 0.204·24-s + 0.200·25-s + 0.192·27-s + 0.688·28-s + 1.86·29-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
\(L(1)\) |
\(\approx\) |
\(2.574231506\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.574231506\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.64T + 7T^{2} \) |
| 11 | \( 1 + 1.66T + 11T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 - 6.31T + 19T^{2} \) |
| 23 | \( 1 - 1.24T + 23T^{2} \) |
| 29 | \( 1 - 10.0T + 29T^{2} \) |
| 31 | \( 1 + 4.21T + 31T^{2} \) |
| 37 | \( 1 - 9.86T + 37T^{2} \) |
| 41 | \( 1 + 9.28T + 41T^{2} \) |
| 43 | \( 1 + 7.57T + 43T^{2} \) |
| 47 | \( 1 + 6.82T + 47T^{2} \) |
| 53 | \( 1 + 0.848T + 53T^{2} \) |
| 59 | \( 1 + 6.10T + 59T^{2} \) |
| 61 | \( 1 - 7.46T + 61T^{2} \) |
| 67 | \( 1 + 14.7T + 67T^{2} \) |
| 71 | \( 1 + 3.51T + 71T^{2} \) |
| 73 | \( 1 - 12.2T + 73T^{2} \) |
| 79 | \( 1 - 9.93T + 79T^{2} \) |
| 83 | \( 1 + 7.95T + 83T^{2} \) |
| 89 | \( 1 - 5.95T + 89T^{2} \) |
| 97 | \( 1 + 2.75T + 97T^{2} \) |
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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
−8.128487437688623352668396681464, −7.81009723469993220255801037418, −7.07952163118730238431864262557, −6.15301807044221794027874438068, −5.17007559805906567136007275107, −4.79571733980393321871939888286, −3.42927013666057917774808813484, −2.73051132904940938365469557864, −1.71454879044491847523544570217, −1.04655167751832239974043186229,
1.04655167751832239974043186229, 1.71454879044491847523544570217, 2.73051132904940938365469557864, 3.42927013666057917774808813484, 4.79571733980393321871939888286, 5.17007559805906567136007275107, 6.15301807044221794027874438068, 7.07952163118730238431864262557, 7.81009723469993220255801037418, 8.128487437688623352668396681464