| L(s) = 1 | + 1.33·3-s − 18.4·5-s − 10.3·7-s − 25.2·9-s + 50.4·11-s + 61.8·13-s − 24.5·15-s + 68.1·17-s − 19·19-s − 13.8·21-s − 145.·23-s + 215.·25-s − 69.5·27-s − 42.6·29-s − 91.6·31-s + 67.1·33-s + 191.·35-s + 400.·37-s + 82.3·39-s − 123.·41-s + 449.·43-s + 465.·45-s + 453.·47-s − 235.·49-s + 90.7·51-s − 437.·53-s − 930.·55-s + ⋯ |
| L(s) = 1 | + 0.256·3-s − 1.64·5-s − 0.560·7-s − 0.934·9-s + 1.38·11-s + 1.31·13-s − 0.422·15-s + 0.971·17-s − 0.229·19-s − 0.143·21-s − 1.32·23-s + 1.72·25-s − 0.495·27-s − 0.272·29-s − 0.531·31-s + 0.354·33-s + 0.925·35-s + 1.78·37-s + 0.338·39-s − 0.469·41-s + 1.59·43-s + 1.54·45-s + 1.40·47-s − 0.685·49-s + 0.249·51-s − 1.13·53-s − 2.28·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + 19T \) |
| good | 3 | \( 1 - 1.33T + 27T^{2} \) |
| 5 | \( 1 + 18.4T + 125T^{2} \) |
| 7 | \( 1 + 10.3T + 343T^{2} \) |
| 11 | \( 1 - 50.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 61.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 68.1T + 4.91e3T^{2} \) |
| 23 | \( 1 + 145.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 42.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 91.6T + 2.97e4T^{2} \) |
| 37 | \( 1 - 400.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 123.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 449.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 453.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 437.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 159.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 476.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 629.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 471.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 725.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.05e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 726.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 468.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 891.T + 9.12e5T^{2} \) |
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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
−8.851791780602419607602219767912, −8.118714847093142627613289334021, −7.52684023407028742104361608125, −6.39101654064133441094071441624, −5.78429799096575595373958250322, −4.11144976458328578199434299967, −3.82238115604235617356894308115, −2.91698014957202778824429916160, −1.17653407856076175370535874376, 0,
1.17653407856076175370535874376, 2.91698014957202778824429916160, 3.82238115604235617356894308115, 4.11144976458328578199434299967, 5.78429799096575595373958250322, 6.39101654064133441094071441624, 7.52684023407028742104361608125, 8.118714847093142627613289334021, 8.851791780602419607602219767912
