| L(s) = 1 | − 2.42·2-s + 1.22·3-s + 3.89·4-s + 0.497·5-s − 2.96·6-s + 7-s − 4.60·8-s − 1.50·9-s − 1.20·10-s + 5.72·11-s + 4.75·12-s + 2.95·13-s − 2.42·14-s + 0.608·15-s + 3.38·16-s + 1.50·17-s + 3.66·18-s + 0.453·19-s + 1.94·20-s + 1.22·21-s − 13.9·22-s − 9.08·23-s − 5.62·24-s − 4.75·25-s − 7.18·26-s − 5.50·27-s + 3.89·28-s + ⋯ |
| L(s) = 1 | − 1.71·2-s + 0.705·3-s + 1.94·4-s + 0.222·5-s − 1.21·6-s + 0.377·7-s − 1.62·8-s − 0.502·9-s − 0.382·10-s + 1.72·11-s + 1.37·12-s + 0.820·13-s − 0.648·14-s + 0.157·15-s + 0.847·16-s + 0.365·17-s + 0.863·18-s + 0.104·19-s + 0.433·20-s + 0.266·21-s − 2.96·22-s − 1.89·23-s − 1.14·24-s − 0.950·25-s − 1.40·26-s − 1.05·27-s + 0.736·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.239052506\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.239052506\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 859 | \( 1 + T \) |
| good | 2 | \( 1 + 2.42T + 2T^{2} \) |
| 3 | \( 1 - 1.22T + 3T^{2} \) |
| 5 | \( 1 - 0.497T + 5T^{2} \) |
| 11 | \( 1 - 5.72T + 11T^{2} \) |
| 13 | \( 1 - 2.95T + 13T^{2} \) |
| 17 | \( 1 - 1.50T + 17T^{2} \) |
| 19 | \( 1 - 0.453T + 19T^{2} \) |
| 23 | \( 1 + 9.08T + 23T^{2} \) |
| 29 | \( 1 + 6.84T + 29T^{2} \) |
| 31 | \( 1 + 9.30T + 31T^{2} \) |
| 37 | \( 1 + 0.592T + 37T^{2} \) |
| 41 | \( 1 - 7.11T + 41T^{2} \) |
| 43 | \( 1 - 7.01T + 43T^{2} \) |
| 47 | \( 1 - 0.677T + 47T^{2} \) |
| 53 | \( 1 - 8.46T + 53T^{2} \) |
| 59 | \( 1 - 5.81T + 59T^{2} \) |
| 61 | \( 1 - 10.5T + 61T^{2} \) |
| 67 | \( 1 - 4.09T + 67T^{2} \) |
| 71 | \( 1 - 1.08T + 71T^{2} \) |
| 73 | \( 1 + 0.421T + 73T^{2} \) |
| 79 | \( 1 - 12.5T + 79T^{2} \) |
| 83 | \( 1 - 6.18T + 83T^{2} \) |
| 89 | \( 1 - 5.14T + 89T^{2} \) |
| 97 | \( 1 + 0.785T + 97T^{2} \) |
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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
−8.216200514624382117153716680571, −7.67356546531882454238777883623, −7.03116128947183462247386550964, −6.03361240044081237620904176578, −5.72319939555586092414462471358, −3.94084186532903100440588161451, −3.66146149627824160919936625839, −2.15099209126464065222230183522, −1.86416973321610091151848235598, −0.73950113000676074605490657799,
0.73950113000676074605490657799, 1.86416973321610091151848235598, 2.15099209126464065222230183522, 3.66146149627824160919936625839, 3.94084186532903100440588161451, 5.72319939555586092414462471358, 6.03361240044081237620904176578, 7.03116128947183462247386550964, 7.67356546531882454238777883623, 8.216200514624382117153716680571
