| L(s) = 1 | − 2.02·3-s − 0.0986·5-s + 4.09·7-s + 1.09·9-s − 0.341·11-s − 1.10·13-s + 0.199·15-s + 5.70·17-s − 1.98·19-s − 8.29·21-s + 5.11·23-s − 4.99·25-s + 3.85·27-s − 8.70·29-s + 2.21·31-s + 0.691·33-s − 0.404·35-s + 2.22·37-s + 2.22·39-s − 2.31·41-s − 0.00233·43-s − 0.107·45-s + 3.66·47-s + 9.80·49-s − 11.5·51-s + 9.27·53-s + 0.0337·55-s + ⋯ |
| L(s) = 1 | − 1.16·3-s − 0.0440·5-s + 1.54·7-s + 0.364·9-s − 0.103·11-s − 0.305·13-s + 0.0515·15-s + 1.38·17-s − 0.455·19-s − 1.81·21-s + 1.06·23-s − 0.998·25-s + 0.741·27-s − 1.61·29-s + 0.397·31-s + 0.120·33-s − 0.0683·35-s + 0.365·37-s + 0.356·39-s − 0.362·41-s − 0.000356·43-s − 0.0160·45-s + 0.534·47-s + 1.40·49-s − 1.61·51-s + 1.27·53-s + 0.00454·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.249510443\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.249510443\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 167 | \( 1 - T \) |
| good | 3 | \( 1 + 2.02T + 3T^{2} \) |
| 5 | \( 1 + 0.0986T + 5T^{2} \) |
| 7 | \( 1 - 4.09T + 7T^{2} \) |
| 11 | \( 1 + 0.341T + 11T^{2} \) |
| 13 | \( 1 + 1.10T + 13T^{2} \) |
| 17 | \( 1 - 5.70T + 17T^{2} \) |
| 19 | \( 1 + 1.98T + 19T^{2} \) |
| 23 | \( 1 - 5.11T + 23T^{2} \) |
| 29 | \( 1 + 8.70T + 29T^{2} \) |
| 31 | \( 1 - 2.21T + 31T^{2} \) |
| 37 | \( 1 - 2.22T + 37T^{2} \) |
| 41 | \( 1 + 2.31T + 41T^{2} \) |
| 43 | \( 1 + 0.00233T + 43T^{2} \) |
| 47 | \( 1 - 3.66T + 47T^{2} \) |
| 53 | \( 1 - 9.27T + 53T^{2} \) |
| 59 | \( 1 - 4.99T + 59T^{2} \) |
| 61 | \( 1 + 0.762T + 61T^{2} \) |
| 67 | \( 1 - 1.26T + 67T^{2} \) |
| 71 | \( 1 - 13.5T + 71T^{2} \) |
| 73 | \( 1 + 7.04T + 73T^{2} \) |
| 79 | \( 1 - 17.0T + 79T^{2} \) |
| 83 | \( 1 - 14.1T + 83T^{2} \) |
| 89 | \( 1 - 0.590T + 89T^{2} \) |
| 97 | \( 1 - 6.64T + 97T^{2} \) |
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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
−9.790126468584960676476727540202, −8.728350070224294931791073655936, −7.86196135918943011635618521464, −7.26717049891019935038830809904, −6.08619882605618287729858753653, −5.33699957841746052312718503325, −4.86851054931178846075839134103, −3.71635652699288376444574889329, −2.14342356436274328223028457879, −0.895928643721691959666107822279,
0.895928643721691959666107822279, 2.14342356436274328223028457879, 3.71635652699288376444574889329, 4.86851054931178846075839134103, 5.33699957841746052312718503325, 6.08619882605618287729858753653, 7.26717049891019935038830809904, 7.86196135918943011635618521464, 8.728350070224294931791073655936, 9.790126468584960676476727540202
