| L(s) = 1 | − 1.61·2-s + 0.618·4-s + 2.23·5-s + 7-s + 2.23·8-s − 3.61·10-s + 0.236·13-s − 1.61·14-s − 4.85·16-s − 2·17-s − 1.47·19-s + 1.38·20-s + 2·23-s − 0.381·26-s + 0.618·28-s − 5·29-s + 10.4·31-s + 3.38·32-s + 3.23·34-s + 2.23·35-s − 1.47·37-s + 2.38·38-s + 5.00·40-s − 2.47·41-s − 8.47·43-s − 3.23·46-s − 9.47·47-s + ⋯ |
| L(s) = 1 | − 1.14·2-s + 0.309·4-s + 0.999·5-s + 0.377·7-s + 0.790·8-s − 1.14·10-s + 0.0654·13-s − 0.432·14-s − 1.21·16-s − 0.485·17-s − 0.337·19-s + 0.309·20-s + 0.417·23-s − 0.0749·26-s + 0.116·28-s − 0.928·29-s + 1.88·31-s + 0.597·32-s + 0.554·34-s + 0.377·35-s − 0.242·37-s + 0.386·38-s + 0.790·40-s − 0.386·41-s − 1.29·43-s − 0.477·46-s − 1.38·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 5 | \( 1 - 2.23T + 5T^{2} \) |
| 13 | \( 1 - 0.236T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 1.47T + 19T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 + 1.47T + 37T^{2} \) |
| 41 | \( 1 + 2.47T + 41T^{2} \) |
| 43 | \( 1 + 8.47T + 43T^{2} \) |
| 47 | \( 1 + 9.47T + 47T^{2} \) |
| 53 | \( 1 + 6.47T + 53T^{2} \) |
| 59 | \( 1 + 7.94T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 - 9.18T + 67T^{2} \) |
| 71 | \( 1 - 2.47T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 + 0.472T + 79T^{2} \) |
| 83 | \( 1 + 7.52T + 83T^{2} \) |
| 89 | \( 1 - 0.472T + 89T^{2} \) |
| 97 | \( 1 - 9.41T + 97T^{2} \) |
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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
−7.83010232123211206288182724240, −6.79197779603329972875071808828, −6.45263885359085466676699885698, −5.39375707161474817378761251087, −4.83081410640440801215156267248, −3.99050152611433470843246374556, −2.79113172341843901723743976346, −1.89498610757765272751121870617, −1.30252169858491426478512536069, 0,
1.30252169858491426478512536069, 1.89498610757765272751121870617, 2.79113172341843901723743976346, 3.99050152611433470843246374556, 4.83081410640440801215156267248, 5.39375707161474817378761251087, 6.45263885359085466676699885698, 6.79197779603329972875071808828, 7.83010232123211206288182724240
