| L(s) = 1 | − 2.51·2-s − 0.754·3-s + 4.31·4-s − 0.589·5-s + 1.89·6-s + 4.62·7-s − 5.81·8-s − 2.43·9-s + 1.48·10-s − 1.16·11-s − 3.25·12-s − 3.22·13-s − 11.6·14-s + 0.445·15-s + 5.98·16-s − 1.25·17-s + 6.10·18-s − 5.66·19-s − 2.54·20-s − 3.48·21-s + 2.93·22-s + 8.36·23-s + 4.38·24-s − 4.65·25-s + 8.11·26-s + 4.09·27-s + 19.9·28-s + ⋯ |
| L(s) = 1 | − 1.77·2-s − 0.435·3-s + 2.15·4-s − 0.263·5-s + 0.774·6-s + 1.74·7-s − 2.05·8-s − 0.810·9-s + 0.468·10-s − 0.352·11-s − 0.940·12-s − 0.895·13-s − 3.10·14-s + 0.114·15-s + 1.49·16-s − 0.305·17-s + 1.43·18-s − 1.29·19-s − 0.568·20-s − 0.761·21-s + 0.626·22-s + 1.74·23-s + 0.895·24-s − 0.930·25-s + 1.59·26-s + 0.788·27-s + 3.76·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9409 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9409 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4804891610\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4804891610\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 97 | \( 1 \) |
| good | 2 | \( 1 + 2.51T + 2T^{2} \) |
| 3 | \( 1 + 0.754T + 3T^{2} \) |
| 5 | \( 1 + 0.589T + 5T^{2} \) |
| 7 | \( 1 - 4.62T + 7T^{2} \) |
| 11 | \( 1 + 1.16T + 11T^{2} \) |
| 13 | \( 1 + 3.22T + 13T^{2} \) |
| 17 | \( 1 + 1.25T + 17T^{2} \) |
| 19 | \( 1 + 5.66T + 19T^{2} \) |
| 23 | \( 1 - 8.36T + 23T^{2} \) |
| 29 | \( 1 - 3.95T + 29T^{2} \) |
| 31 | \( 1 + 0.301T + 31T^{2} \) |
| 37 | \( 1 - 5.09T + 37T^{2} \) |
| 41 | \( 1 + 3.74T + 41T^{2} \) |
| 43 | \( 1 + 11.3T + 43T^{2} \) |
| 47 | \( 1 - 8.24T + 47T^{2} \) |
| 53 | \( 1 + 1.61T + 53T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 + 4.00T + 61T^{2} \) |
| 67 | \( 1 + 2.02T + 67T^{2} \) |
| 71 | \( 1 - 13.7T + 71T^{2} \) |
| 73 | \( 1 + 9.86T + 73T^{2} \) |
| 79 | \( 1 + 5.68T + 79T^{2} \) |
| 83 | \( 1 + 2.51T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
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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
−7.892055777900003230822459326532, −7.30663922067684244685623336320, −6.64562670565765798459711978456, −5.81769391683522201806184564473, −4.95181686357708706707559740445, −4.48451790448486749352857175370, −2.95970523891491765930275886075, −2.25129649793036909992956772048, −1.50053569542823725053405426063, −0.45452436033964168463705656110,
0.45452436033964168463705656110, 1.50053569542823725053405426063, 2.25129649793036909992956772048, 2.95970523891491765930275886075, 4.48451790448486749352857175370, 4.95181686357708706707559740445, 5.81769391683522201806184564473, 6.64562670565765798459711978456, 7.30663922067684244685623336320, 7.892055777900003230822459326532
