| L(s) = 1 | + 1.17·2-s − 3.34·3-s − 0.629·4-s + 0.135·5-s − 3.91·6-s − 3.07·8-s + 8.21·9-s + 0.158·10-s + 4.30·11-s + 2.10·12-s + 1.09·13-s − 0.452·15-s − 2.34·16-s − 4.41·17-s + 9.61·18-s + 2.11·19-s − 0.0850·20-s + 5.03·22-s + 23-s + 10.3·24-s − 4.98·25-s + 1.27·26-s − 17.4·27-s − 2.57·29-s − 0.529·30-s + 2.12·31-s + 3.41·32-s + ⋯ |
| L(s) = 1 | + 0.827·2-s − 1.93·3-s − 0.314·4-s + 0.0603·5-s − 1.60·6-s − 1.08·8-s + 2.73·9-s + 0.0499·10-s + 1.29·11-s + 0.608·12-s + 0.303·13-s − 0.116·15-s − 0.585·16-s − 1.07·17-s + 2.26·18-s + 0.484·19-s − 0.0190·20-s + 1.07·22-s + 0.208·23-s + 2.10·24-s − 0.996·25-s + 0.250·26-s − 3.35·27-s − 0.477·29-s − 0.0966·30-s + 0.382·31-s + 0.603·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1127 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - 1.17T + 2T^{2} \) |
| 3 | \( 1 + 3.34T + 3T^{2} \) |
| 5 | \( 1 - 0.135T + 5T^{2} \) |
| 11 | \( 1 - 4.30T + 11T^{2} \) |
| 13 | \( 1 - 1.09T + 13T^{2} \) |
| 17 | \( 1 + 4.41T + 17T^{2} \) |
| 19 | \( 1 - 2.11T + 19T^{2} \) |
| 29 | \( 1 + 2.57T + 29T^{2} \) |
| 31 | \( 1 - 2.12T + 31T^{2} \) |
| 37 | \( 1 + 10.7T + 37T^{2} \) |
| 41 | \( 1 + 4.16T + 41T^{2} \) |
| 43 | \( 1 - 0.313T + 43T^{2} \) |
| 47 | \( 1 - 0.456T + 47T^{2} \) |
| 53 | \( 1 + 1.47T + 53T^{2} \) |
| 59 | \( 1 + 12.3T + 59T^{2} \) |
| 61 | \( 1 - 4.69T + 61T^{2} \) |
| 67 | \( 1 + 8.88T + 67T^{2} \) |
| 71 | \( 1 + 5.37T + 71T^{2} \) |
| 73 | \( 1 + 12.6T + 73T^{2} \) |
| 79 | \( 1 + 3.64T + 79T^{2} \) |
| 83 | \( 1 - 9.83T + 83T^{2} \) |
| 89 | \( 1 + 0.699T + 89T^{2} \) |
| 97 | \( 1 - 8.12T + 97T^{2} \) |
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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
−9.556733890053133818890196219332, −8.815227633437180178019577941756, −7.28384100317432774652981521409, −6.45292117993928482831659420973, −5.96933003571840260144989967010, −5.10448026226987566310201231649, −4.39094403571882965571487313731, −3.64100492457981779933397302624, −1.51478973835796430656038139466, 0,
1.51478973835796430656038139466, 3.64100492457981779933397302624, 4.39094403571882965571487313731, 5.10448026226987566310201231649, 5.96933003571840260144989967010, 6.45292117993928482831659420973, 7.28384100317432774652981521409, 8.815227633437180178019577941756, 9.556733890053133818890196219332
