| L(s) = 1 | + 1.02e3·4-s + 5.90e4·9-s − 1.85e4·11-s + 3.03e5·13-s + 1.04e6·16-s − 2.76e6·17-s + 4.12e6·23-s + 9.76e6·25-s + 5.72e7·31-s + 6.04e7·36-s + 1.42e8·41-s − 1.47e8·43-s − 1.89e7·44-s + 4.51e8·47-s + 2.82e8·49-s + 3.11e8·52-s − 3.39e7·53-s − 9.90e8·59-s + 1.07e9·64-s − 1.50e9·67-s − 2.83e9·68-s − 2.64e9·79-s + 3.48e9·81-s − 6.75e9·83-s + 4.22e9·92-s − 1.50e10·97-s − 1.09e9·99-s + ⋯ |
| L(s) = 1 | + 4-s + 9-s − 0.114·11-s + 0.818·13-s + 16-s − 1.94·17-s + 0.641·23-s + 25-s + 1.99·31-s + 36-s + 1.23·41-s − 43-s − 0.114·44-s + 1.96·47-s + 49-s + 0.818·52-s − 0.0812·53-s − 1.38·59-s + 64-s − 1.11·67-s − 1.94·68-s − 0.858·79-s + 81-s − 1.71·83-s + 0.641·92-s − 1.74·97-s − 0.114·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(2.809697913\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.809697913\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + p^{5} T \) |
| good | 2 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 3 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 5 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 7 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 11 | \( 1 + 18501 T + p^{10} T^{2} \) |
| 13 | \( 1 - 303943 T + p^{10} T^{2} \) |
| 17 | \( 1 + 2764089 T + p^{10} T^{2} \) |
| 19 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 23 | \( 1 - 4126443 T + p^{10} T^{2} \) |
| 29 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 31 | \( 1 - 57253099 T + p^{10} T^{2} \) |
| 37 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 41 | \( 1 - 142671399 T + p^{10} T^{2} \) |
| 47 | \( 1 - 451176882 T + p^{10} T^{2} \) |
| 53 | \( 1 + 33972057 T + p^{10} T^{2} \) |
| 59 | \( 1 + 990191574 T + p^{10} T^{2} \) |
| 61 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 67 | \( 1 + 1504819589 T + p^{10} T^{2} \) |
| 71 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 73 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 79 | \( 1 + 2641416974 T + p^{10} T^{2} \) |
| 83 | \( 1 + 6757639557 T + p^{10} T^{2} \) |
| 89 | \( ( 1 - p^{5} T )( 1 + p^{5} T ) \) |
| 97 | \( 1 + 15006753793 T + p^{10} T^{2} \) |
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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
−13.61968985847640932962343317376, −12.55575504167204922346203540064, −11.23475373402581615493898358447, −10.38750795901089135860605357578, −8.766017487478642134785561873987, −7.22161228931604059319959815758, −6.28754246321329285837254271214, −4.39527417219450588265669809377, −2.62727087722162381857371365653, −1.17894872103903504229698025076,
1.17894872103903504229698025076, 2.62727087722162381857371365653, 4.39527417219450588265669809377, 6.28754246321329285837254271214, 7.22161228931604059319959815758, 8.766017487478642134785561873987, 10.38750795901089135860605357578, 11.23475373402581615493898358447, 12.55575504167204922346203540064, 13.61968985847640932962343317376
