L(s) = 1 | + 2·3-s + 2·7-s + 9-s − 5·11-s − 6·13-s + 7·19-s + 4·21-s + 2·23-s − 4·27-s + 9·29-s − 8·31-s − 10·33-s − 12·39-s − 7·41-s + 10·43-s + 8·47-s − 3·49-s + 4·53-s + 14·57-s + 9·59-s + 5·61-s + 2·63-s + 4·67-s + 4·69-s − 71-s − 4·73-s − 10·77-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 0.755·7-s + 1/3·9-s − 1.50·11-s − 1.66·13-s + 1.60·19-s + 0.872·21-s + 0.417·23-s − 0.769·27-s + 1.67·29-s − 1.43·31-s − 1.74·33-s − 1.92·39-s − 1.09·41-s + 1.52·43-s + 1.16·47-s − 3/7·49-s + 0.549·53-s + 1.85·57-s + 1.17·59-s + 0.640·61-s + 0.251·63-s + 0.488·67-s + 0.481·69-s − 0.118·71-s − 0.468·73-s − 1.13·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 17 | \( 1 \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−15.32636049173909, −14.85520671266958, −14.23841693208610, −14.12008152947377, −13.43142596173638, −12.88348148422447, −12.31429000114887, −11.77247497596427, −11.14182645744634, −10.52358791409601, −9.843632953297810, −9.624382664039723, −8.747768860147224, −8.370277839628524, −7.754400742789737, −7.326381123895950, −7.002967284548155, −5.548538603278196, −5.402736636235321, −4.744639676894113, −4.001680377769927, −3.053065592357211, −2.669633884753383, −2.194840708588074, −1.171984512583126, 0,
1.171984512583126, 2.194840708588074, 2.669633884753383, 3.053065592357211, 4.001680377769927, 4.744639676894113, 5.402736636235321, 5.548538603278196, 7.002967284548155, 7.326381123895950, 7.754400742789737, 8.370277839628524, 8.747768860147224, 9.624382664039723, 9.843632953297810, 10.52358791409601, 11.14182645744634, 11.77247497596427, 12.31429000114887, 12.88348148422447, 13.43142596173638, 14.12008152947377, 14.23841693208610, 14.85520671266958, 15.32636049173909