| L(s) = 1 | + 2·2-s + 3-s + 2·4-s − 4·5-s + 2·6-s + 2·7-s + 9-s − 8·10-s + 2·12-s + 6·13-s + 4·14-s − 4·15-s − 4·16-s − 3·17-s + 2·18-s − 8·20-s + 2·21-s − 6·23-s + 11·25-s + 12·26-s + 27-s + 4·28-s − 5·29-s − 8·30-s + 7·31-s − 8·32-s − 6·34-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s − 1.78·5-s + 0.816·6-s + 0.755·7-s + 1/3·9-s − 2.52·10-s + 0.577·12-s + 1.66·13-s + 1.06·14-s − 1.03·15-s − 16-s − 0.727·17-s + 0.471·18-s − 1.78·20-s + 0.436·21-s − 1.25·23-s + 11/5·25-s + 2.35·26-s + 0.192·27-s + 0.755·28-s − 0.928·29-s − 1.46·30-s + 1.25·31-s − 1.41·32-s − 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14883 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14883 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 11 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−15.95858952160795, −15.55796018980428, −15.25338335232990, −14.69258529797526, −13.98631640575535, −13.75583483958093, −13.03188712596971, −12.48655470177139, −11.88212722104592, −11.46732611567406, −11.03634482799982, −10.43420989470370, −9.261012391969417, −8.636121671628144, −8.180365905128470, −7.753695514112764, −6.866759360685973, −6.382330447041978, −5.520118436026387, −4.721075821120553, −4.207851291757509, −3.755202985105525, −3.317168663095771, −2.375434982184406, −1.362017472103255, 0,
1.362017472103255, 2.375434982184406, 3.317168663095771, 3.755202985105525, 4.207851291757509, 4.721075821120553, 5.520118436026387, 6.382330447041978, 6.866759360685973, 7.753695514112764, 8.180365905128470, 8.636121671628144, 9.261012391969417, 10.43420989470370, 11.03634482799982, 11.46732611567406, 11.88212722104592, 12.48655470177139, 13.03188712596971, 13.75583483958093, 13.98631640575535, 14.69258529797526, 15.25338335232990, 15.55796018980428, 15.95858952160795
