| L(s) = 1 | − 2.56·3-s − 2·5-s + 3.56·9-s + 5.12·11-s − 4.56·13-s + 5.12·15-s − 3.12·17-s + 5.12·19-s + 23-s − 25-s − 1.43·27-s + 0.561·29-s + 6.56·31-s − 13.1·33-s + 8.24·37-s + 11.6·39-s + 10.8·41-s − 8·43-s − 7.12·45-s − 11.6·47-s − 7·49-s + 8·51-s − 2·53-s − 10.2·55-s − 13.1·57-s − 6.24·59-s − 12.2·61-s + ⋯ |
| L(s) = 1 | − 1.47·3-s − 0.894·5-s + 1.18·9-s + 1.54·11-s − 1.26·13-s + 1.32·15-s − 0.757·17-s + 1.17·19-s + 0.208·23-s − 0.200·25-s − 0.276·27-s + 0.104·29-s + 1.17·31-s − 2.28·33-s + 1.35·37-s + 1.87·39-s + 1.68·41-s − 1.21·43-s − 1.06·45-s − 1.70·47-s − 49-s + 1.12·51-s − 0.274·53-s − 1.38·55-s − 1.73·57-s − 0.813·59-s − 1.56·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 2.56T + 3T^{2} \) |
| 5 | \( 1 + 2T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 5.12T + 11T^{2} \) |
| 13 | \( 1 + 4.56T + 13T^{2} \) |
| 17 | \( 1 + 3.12T + 17T^{2} \) |
| 19 | \( 1 - 5.12T + 19T^{2} \) |
| 29 | \( 1 - 0.561T + 29T^{2} \) |
| 31 | \( 1 - 6.56T + 31T^{2} \) |
| 37 | \( 1 - 8.24T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 6.24T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 + 5.12T + 67T^{2} \) |
| 71 | \( 1 + 9.43T + 71T^{2} \) |
| 73 | \( 1 + 2.31T + 73T^{2} \) |
| 79 | \( 1 - 5.12T + 79T^{2} \) |
| 83 | \( 1 + 2.24T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 + 13.3T + 97T^{2} \) |
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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
−9.382849150871586848509360896520, −8.132115689447926497710853094440, −7.30057266574652109311325660901, −6.59242096916300090184090663137, −5.91532066573763485954839514580, −4.71150820830754754565370046171, −4.39411812799979121380234921574, −3.06055617911215464314095458708, −1.28794341500950429886473213746, 0,
1.28794341500950429886473213746, 3.06055617911215464314095458708, 4.39411812799979121380234921574, 4.71150820830754754565370046171, 5.91532066573763485954839514580, 6.59242096916300090184090663137, 7.30057266574652109311325660901, 8.132115689447926497710853094440, 9.382849150871586848509360896520
