| L(s) = 1 | − 3-s − 3.45·5-s + 2.53·7-s + 9-s + 3.13·11-s + 1.30·13-s + 3.45·15-s + 0.691·17-s − 5.13·19-s − 2.53·21-s − 3.13·23-s + 6.91·25-s − 27-s + 1.75·29-s − 3.40·31-s − 3.13·33-s − 8.75·35-s + 1.62·37-s − 1.30·39-s + 4.46·41-s − 9.31·43-s − 3.45·45-s − 3.15·47-s − 0.568·49-s − 0.691·51-s + 4.42·53-s − 10.8·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.54·5-s + 0.958·7-s + 0.333·9-s + 0.945·11-s + 0.362·13-s + 0.891·15-s + 0.167·17-s − 1.17·19-s − 0.553·21-s − 0.653·23-s + 1.38·25-s − 0.192·27-s + 0.325·29-s − 0.610·31-s − 0.545·33-s − 1.47·35-s + 0.267·37-s − 0.209·39-s + 0.696·41-s − 1.42·43-s − 0.514·45-s − 0.459·47-s − 0.0811·49-s − 0.0967·51-s + 0.607·53-s − 1.45·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 + 3.45T + 5T^{2} \) |
| 7 | \( 1 - 2.53T + 7T^{2} \) |
| 11 | \( 1 - 3.13T + 11T^{2} \) |
| 13 | \( 1 - 1.30T + 13T^{2} \) |
| 17 | \( 1 - 0.691T + 17T^{2} \) |
| 19 | \( 1 + 5.13T + 19T^{2} \) |
| 23 | \( 1 + 3.13T + 23T^{2} \) |
| 29 | \( 1 - 1.75T + 29T^{2} \) |
| 31 | \( 1 + 3.40T + 31T^{2} \) |
| 37 | \( 1 - 1.62T + 37T^{2} \) |
| 41 | \( 1 - 4.46T + 41T^{2} \) |
| 43 | \( 1 + 9.31T + 43T^{2} \) |
| 47 | \( 1 + 3.15T + 47T^{2} \) |
| 53 | \( 1 - 4.42T + 53T^{2} \) |
| 59 | \( 1 - 0.410T + 59T^{2} \) |
| 61 | \( 1 + 5.68T + 61T^{2} \) |
| 67 | \( 1 + 8.25T + 67T^{2} \) |
| 71 | \( 1 - 14.2T + 71T^{2} \) |
| 73 | \( 1 - 1.03T + 73T^{2} \) |
| 79 | \( 1 - 2.46T + 79T^{2} \) |
| 83 | \( 1 - 14.0T + 83T^{2} \) |
| 89 | \( 1 - 12.2T + 89T^{2} \) |
| 97 | \( 1 + 3.13T + 97T^{2} \) |
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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
−7.64805457924042347322525463623, −6.73204662882308768972846314414, −6.28183538010452824577506667475, −5.24759407576890689817620610748, −4.53637082553685938691994057766, −4.02822315538977082946708610615, −3.42287636059051352020182597715, −2.05296135066200825623659569026, −1.11101056466128181940294331970, 0,
1.11101056466128181940294331970, 2.05296135066200825623659569026, 3.42287636059051352020182597715, 4.02822315538977082946708610615, 4.53637082553685938691994057766, 5.24759407576890689817620610748, 6.28183538010452824577506667475, 6.73204662882308768972846314414, 7.64805457924042347322525463623
