| L(s) = 1 | − 4.42·2-s + 11.5·4-s + 5·5-s − 15.6·8-s − 22.1·10-s − 41.6·11-s + 78.9·13-s − 23.1·16-s + 34.5·17-s − 31.5·19-s + 57.7·20-s + 184.·22-s − 3.05·23-s + 25·25-s − 348.·26-s − 87.7·29-s − 274.·31-s + 227.·32-s − 152.·34-s + 283.·37-s + 139.·38-s − 78.3·40-s + 57.3·41-s + 502.·43-s − 481.·44-s + 13.5·46-s − 470.·47-s + ⋯ |
| L(s) = 1 | − 1.56·2-s + 1.44·4-s + 0.447·5-s − 0.692·8-s − 0.698·10-s − 1.14·11-s + 1.68·13-s − 0.361·16-s + 0.493·17-s − 0.380·19-s + 0.645·20-s + 1.78·22-s − 0.0276·23-s + 0.200·25-s − 2.63·26-s − 0.561·29-s − 1.59·31-s + 1.25·32-s − 0.771·34-s + 1.26·37-s + 0.594·38-s − 0.309·40-s + 0.218·41-s + 1.78·43-s − 1.64·44-s + 0.0432·46-s − 1.45·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 4.42T + 8T^{2} \) |
| 11 | \( 1 + 41.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 78.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 34.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 31.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 3.05T + 1.21e4T^{2} \) |
| 29 | \( 1 + 87.7T + 2.43e4T^{2} \) |
| 31 | \( 1 + 274.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 283.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 57.3T + 6.89e4T^{2} \) |
| 43 | \( 1 - 502.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 470.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 682.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 555.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 160.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 67.2T + 3.00e5T^{2} \) |
| 71 | \( 1 - 221.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 237.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 475.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.15e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 924.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 144.T + 9.12e5T^{2} \) |
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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
−8.246288750190714470017328305998, −7.915226512757158615776016426055, −7.00722660815664805592889213196, −6.10846059032010136672120903383, −5.44146864511414001483285723610, −4.14980430650487281841646159797, −2.97592188290519454194950126049, −1.94663301015550546629634456005, −1.10860958443665828161552023685, 0,
1.10860958443665828161552023685, 1.94663301015550546629634456005, 2.97592188290519454194950126049, 4.14980430650487281841646159797, 5.44146864511414001483285723610, 6.10846059032010136672120903383, 7.00722660815664805592889213196, 7.915226512757158615776016426055, 8.246288750190714470017328305998
