| L(s) = 1 | + 2.08·2-s + 2.34·4-s + 0.724·5-s + 1.22·7-s + 0.724·8-s + 1.50·10-s + 1.83·11-s + 3.69·13-s + 2.55·14-s − 3.18·16-s + 6.64·17-s + 1.69·20-s + 3.82·22-s − 3.19·23-s − 4.47·25-s + 7.70·26-s + 2.87·28-s + 5.02·29-s + 2.81·31-s − 8.08·32-s + 13.8·34-s + 0.888·35-s + 7.41·37-s + 0.524·40-s − 10.0·41-s − 0.0641·43-s + 4.30·44-s + ⋯ |
| L(s) = 1 | + 1.47·2-s + 1.17·4-s + 0.323·5-s + 0.463·7-s + 0.256·8-s + 0.477·10-s + 0.552·11-s + 1.02·13-s + 0.683·14-s − 0.796·16-s + 1.61·17-s + 0.380·20-s + 0.815·22-s − 0.666·23-s − 0.895·25-s + 1.51·26-s + 0.544·28-s + 0.933·29-s + 0.505·31-s − 1.42·32-s + 2.37·34-s + 0.150·35-s + 1.21·37-s + 0.0829·40-s − 1.57·41-s − 0.00978·43-s + 0.648·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.161263541\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.161263541\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 2.08T + 2T^{2} \) |
| 5 | \( 1 - 0.724T + 5T^{2} \) |
| 7 | \( 1 - 1.22T + 7T^{2} \) |
| 11 | \( 1 - 1.83T + 11T^{2} \) |
| 13 | \( 1 - 3.69T + 13T^{2} \) |
| 17 | \( 1 - 6.64T + 17T^{2} \) |
| 23 | \( 1 + 3.19T + 23T^{2} \) |
| 29 | \( 1 - 5.02T + 29T^{2} \) |
| 31 | \( 1 - 2.81T + 31T^{2} \) |
| 37 | \( 1 - 7.41T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 + 0.0641T + 43T^{2} \) |
| 47 | \( 1 + 9.67T + 47T^{2} \) |
| 53 | \( 1 - 4.80T + 53T^{2} \) |
| 59 | \( 1 - 7.58T + 59T^{2} \) |
| 61 | \( 1 - 4.17T + 61T^{2} \) |
| 67 | \( 1 - 7.51T + 67T^{2} \) |
| 71 | \( 1 - 14.8T + 71T^{2} \) |
| 73 | \( 1 - 3.90T + 73T^{2} \) |
| 79 | \( 1 - 7.61T + 79T^{2} \) |
| 83 | \( 1 - 12.3T + 83T^{2} \) |
| 89 | \( 1 + 13.6T + 89T^{2} \) |
| 97 | \( 1 - 4.10T + 97T^{2} \) |
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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
−8.388799271297637111255401850291, −7.963075048188142477674139081571, −6.70454232798717807746016950550, −6.24816854407403070228145503722, −5.48333671865334751660796287417, −4.88234331131211297460057313339, −3.86673614355186521370189387729, −3.44071474527569734773622940819, −2.29678294646608638379459107043, −1.21502914976792290507860046469,
1.21502914976792290507860046469, 2.29678294646608638379459107043, 3.44071474527569734773622940819, 3.86673614355186521370189387729, 4.88234331131211297460057313339, 5.48333671865334751660796287417, 6.24816854407403070228145503722, 6.70454232798717807746016950550, 7.963075048188142477674139081571, 8.388799271297637111255401850291
