| L(s) = 1 | + 3.12·3-s − 4.35·5-s + 2.65·7-s + 6.78·9-s − 2.35·11-s + 3.78·13-s − 13.6·15-s − 0.656·17-s + 0.301·19-s + 8.30·21-s + 23-s + 13.9·25-s + 11.8·27-s + 8.49·29-s + 6.52·31-s − 7.36·33-s − 11.5·35-s − 4.35·37-s + 11.8·39-s + 3.07·41-s + 1.44·43-s − 29.5·45-s + 3.73·47-s + 0.0569·49-s − 2.05·51-s + 3.95·53-s + 10.2·55-s + ⋯ |
| L(s) = 1 | + 1.80·3-s − 1.94·5-s + 1.00·7-s + 2.26·9-s − 0.710·11-s + 1.04·13-s − 3.51·15-s − 0.159·17-s + 0.0691·19-s + 1.81·21-s + 0.208·23-s + 2.79·25-s + 2.27·27-s + 1.57·29-s + 1.17·31-s − 1.28·33-s − 1.95·35-s − 0.715·37-s + 1.89·39-s + 0.480·41-s + 0.219·43-s − 4.40·45-s + 0.544·47-s + 0.00813·49-s − 0.287·51-s + 0.543·53-s + 1.38·55-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
| \(L(1)\) |
\(\approx\) |
\(2.691741132\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.691741132\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.12T + 3T^{2} \) |
| 5 | \( 1 + 4.35T + 5T^{2} \) |
| 7 | \( 1 - 2.65T + 7T^{2} \) |
| 11 | \( 1 + 2.35T + 11T^{2} \) |
| 13 | \( 1 - 3.78T + 13T^{2} \) |
| 17 | \( 1 + 0.656T + 17T^{2} \) |
| 19 | \( 1 - 0.301T + 19T^{2} \) |
| 29 | \( 1 - 8.49T + 29T^{2} \) |
| 31 | \( 1 - 6.52T + 31T^{2} \) |
| 37 | \( 1 + 4.35T + 37T^{2} \) |
| 41 | \( 1 - 3.07T + 41T^{2} \) |
| 43 | \( 1 - 1.44T + 43T^{2} \) |
| 47 | \( 1 - 3.73T + 47T^{2} \) |
| 53 | \( 1 - 3.95T + 53T^{2} \) |
| 59 | \( 1 + 13.8T + 59T^{2} \) |
| 61 | \( 1 + 7.95T + 61T^{2} \) |
| 67 | \( 1 - 7.66T + 67T^{2} \) |
| 71 | \( 1 + 13.1T + 71T^{2} \) |
| 73 | \( 1 + 5.33T + 73T^{2} \) |
| 79 | \( 1 - 6.85T + 79T^{2} \) |
| 83 | \( 1 - 8.61T + 83T^{2} \) |
| 89 | \( 1 - 13.5T + 89T^{2} \) |
| 97 | \( 1 - 7.28T + 97T^{2} \) |
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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
−8.999799931355095340703586911523, −8.546958049180224344158422723790, −7.898392316052066499453171610211, −7.68041022416575134150304243267, −6.64478570879382101402787287801, −4.80071089363618672858111756937, −4.26435747268438194608154538646, −3.39622209124435133653986484285, −2.67752824536817928487522418087, −1.18641710904057080086917173502,
1.18641710904057080086917173502, 2.67752824536817928487522418087, 3.39622209124435133653986484285, 4.26435747268438194608154538646, 4.80071089363618672858111756937, 6.64478570879382101402787287801, 7.68041022416575134150304243267, 7.898392316052066499453171610211, 8.546958049180224344158422723790, 8.999799931355095340703586911523
