| L(s) = 1 | − 3·3-s + 5·5-s − 20.3·7-s + 9·9-s + 8.68·11-s − 18·13-s − 15·15-s − 11.7·17-s + 52.6·19-s + 61.0·21-s + 23·23-s + 25·25-s − 27·27-s + 66.4·29-s + 285.·31-s − 26.0·33-s − 101.·35-s − 167.·37-s + 54·39-s + 3.02·41-s − 376.·43-s + 45·45-s − 72.2·47-s + 70.8·49-s + 35.1·51-s + 135.·53-s + 43.4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.09·7-s + 0.333·9-s + 0.238·11-s − 0.384·13-s − 0.258·15-s − 0.167·17-s + 0.636·19-s + 0.634·21-s + 0.208·23-s + 0.200·25-s − 0.192·27-s + 0.425·29-s + 1.65·31-s − 0.137·33-s − 0.491·35-s − 0.745·37-s + 0.221·39-s + 0.0115·41-s − 1.33·43-s + 0.149·45-s − 0.224·47-s + 0.206·49-s + 0.0964·51-s + 0.350·53-s + 0.106·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 5 | \( 1 - 5T \) |
| 23 | \( 1 - 23T \) |
| good | 7 | \( 1 + 20.3T + 343T^{2} \) |
| 11 | \( 1 - 8.68T + 1.33e3T^{2} \) |
| 13 | \( 1 + 18T + 2.19e3T^{2} \) |
| 17 | \( 1 + 11.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 52.6T + 6.85e3T^{2} \) |
| 29 | \( 1 - 66.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 285.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 167.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 3.02T + 6.89e4T^{2} \) |
| 43 | \( 1 + 376.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 72.2T + 1.03e5T^{2} \) |
| 53 | \( 1 - 135.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 526.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 860.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 264.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.02e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 802.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 736.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 240.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 639.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 365.T + 9.12e5T^{2} \) |
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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
−8.980488951760328470782788043030, −7.975033115635418929744091832505, −6.80354548756787768715273027869, −6.50403621805162781525880408192, −5.50524431538655713633026799799, −4.70405886947576682746944494235, −3.52797021981414431137736086239, −2.58727878578631319327944414823, −1.20475456949099819564682650100, 0,
1.20475456949099819564682650100, 2.58727878578631319327944414823, 3.52797021981414431137736086239, 4.70405886947576682746944494235, 5.50524431538655713633026799799, 6.50403621805162781525880408192, 6.80354548756787768715273027869, 7.975033115635418929744091832505, 8.980488951760328470782788043030
