| L(s) = 1 | + 0.353·3-s + 5-s + 1.60·7-s − 2.87·9-s − 5.86·11-s − 0.686·13-s + 0.353·15-s − 0.0112·17-s + 5.24·19-s + 0.566·21-s + 3.26·23-s + 25-s − 2.07·27-s + 1.70·29-s + 3.06·31-s − 2.07·33-s + 1.60·35-s + 10.1·37-s − 0.242·39-s − 0.405·41-s − 4.86·43-s − 2.87·45-s + 7.29·47-s − 4.43·49-s − 0.00399·51-s − 8.31·53-s − 5.86·55-s + ⋯ |
| L(s) = 1 | + 0.204·3-s + 0.447·5-s + 0.605·7-s − 0.958·9-s − 1.76·11-s − 0.190·13-s + 0.0913·15-s − 0.00273·17-s + 1.20·19-s + 0.123·21-s + 0.681·23-s + 0.200·25-s − 0.400·27-s + 0.316·29-s + 0.550·31-s − 0.361·33-s + 0.270·35-s + 1.67·37-s − 0.0388·39-s − 0.0632·41-s − 0.741·43-s − 0.428·45-s + 1.06·47-s − 0.633·49-s − 0.000559·51-s − 1.14·53-s − 0.790·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.008997350\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.008997350\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 3 | \( 1 - 0.353T + 3T^{2} \) |
| 7 | \( 1 - 1.60T + 7T^{2} \) |
| 11 | \( 1 + 5.86T + 11T^{2} \) |
| 13 | \( 1 + 0.686T + 13T^{2} \) |
| 17 | \( 1 + 0.0112T + 17T^{2} \) |
| 19 | \( 1 - 5.24T + 19T^{2} \) |
| 23 | \( 1 - 3.26T + 23T^{2} \) |
| 29 | \( 1 - 1.70T + 29T^{2} \) |
| 31 | \( 1 - 3.06T + 31T^{2} \) |
| 37 | \( 1 - 10.1T + 37T^{2} \) |
| 41 | \( 1 + 0.405T + 41T^{2} \) |
| 43 | \( 1 + 4.86T + 43T^{2} \) |
| 47 | \( 1 - 7.29T + 47T^{2} \) |
| 53 | \( 1 + 8.31T + 53T^{2} \) |
| 59 | \( 1 - 3.40T + 59T^{2} \) |
| 61 | \( 1 + 5.96T + 61T^{2} \) |
| 67 | \( 1 + 6.10T + 67T^{2} \) |
| 71 | \( 1 + 5.70T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 - 3.61T + 79T^{2} \) |
| 83 | \( 1 - 9.53T + 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 - 6.80T + 97T^{2} \) |
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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
−7.82271399879546580125725290395, −7.41550364646151342643188886049, −6.31509985183287620983972540046, −5.64806293148767433849113460758, −5.08434667730414141139265943378, −4.54236100291469354128349042464, −3.09495342760383448499556049440, −2.84123998590880918656785729232, −1.90366874483580387759405836632, −0.67782908549584891915450304163,
0.67782908549584891915450304163, 1.90366874483580387759405836632, 2.84123998590880918656785729232, 3.09495342760383448499556049440, 4.54236100291469354128349042464, 5.08434667730414141139265943378, 5.64806293148767433849113460758, 6.31509985183287620983972540046, 7.41550364646151342643188886049, 7.82271399879546580125725290395
