| L(s) = 1 | − 0.772·2-s − 1.40·4-s − 5-s − 2.17·7-s + 2.62·8-s + 0.772·10-s − 3.40·11-s + 5·13-s + 1.68·14-s + 0.772·16-s − 8.03·17-s − 1.22·19-s + 1.40·20-s + 2.62·22-s − 2.17·23-s + 25-s − 3.86·26-s + 3.05·28-s + 2.40·29-s − 2.17·31-s − 5.85·32-s + 6.20·34-s + 2.17·35-s − 37-s + 0.948·38-s − 2.62·40-s − 1.72·41-s + ⋯ |
| L(s) = 1 | − 0.546·2-s − 0.701·4-s − 0.447·5-s − 0.822·7-s + 0.929·8-s + 0.244·10-s − 1.02·11-s + 1.38·13-s + 0.449·14-s + 0.193·16-s − 1.94·17-s − 0.281·19-s + 0.313·20-s + 0.560·22-s − 0.453·23-s + 0.200·25-s − 0.757·26-s + 0.576·28-s + 0.446·29-s − 0.390·31-s − 1.03·32-s + 1.06·34-s + 0.367·35-s − 0.164·37-s + 0.153·38-s − 0.415·40-s − 0.268·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1665 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1665 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5551885857\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5551885857\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| good | 2 | \( 1 + 0.772T + 2T^{2} \) |
| 7 | \( 1 + 2.17T + 7T^{2} \) |
| 11 | \( 1 + 3.40T + 11T^{2} \) |
| 13 | \( 1 - 5T + 13T^{2} \) |
| 17 | \( 1 + 8.03T + 17T^{2} \) |
| 19 | \( 1 + 1.22T + 19T^{2} \) |
| 23 | \( 1 + 2.17T + 23T^{2} \) |
| 29 | \( 1 - 2.40T + 29T^{2} \) |
| 31 | \( 1 + 2.17T + 31T^{2} \) |
| 41 | \( 1 + 1.72T + 41T^{2} \) |
| 43 | \( 1 - 4.40T + 43T^{2} \) |
| 47 | \( 1 - 2.54T + 47T^{2} \) |
| 53 | \( 1 + 6.08T + 53T^{2} \) |
| 59 | \( 1 - 13.6T + 59T^{2} \) |
| 61 | \( 1 - 11.4T + 61T^{2} \) |
| 67 | \( 1 - 15.2T + 67T^{2} \) |
| 71 | \( 1 + 5.12T + 71T^{2} \) |
| 73 | \( 1 + 0.765T + 73T^{2} \) |
| 79 | \( 1 - 4.66T + 79T^{2} \) |
| 83 | \( 1 - 15.8T + 83T^{2} \) |
| 89 | \( 1 + 0.194T + 89T^{2} \) |
| 97 | \( 1 + 0.980T + 97T^{2} \) |
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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
−9.236515071878924319330027432880, −8.551575441171495358413786366960, −8.103164279554772820816840066390, −7.02144847563449942805346609751, −6.27440839932868382608572220218, −5.20796997994571365788646864192, −4.24788799223300746183720871555, −3.53758345622370783364125525794, −2.20330921543342143454182528533, −0.54626676409861452076001077750,
0.54626676409861452076001077750, 2.20330921543342143454182528533, 3.53758345622370783364125525794, 4.24788799223300746183720871555, 5.20796997994571365788646864192, 6.27440839932868382608572220218, 7.02144847563449942805346609751, 8.103164279554772820816840066390, 8.551575441171495358413786366960, 9.236515071878924319330027432880
