L(s) = 1 | − 3-s − 1.47·5-s + 7-s + 9-s + 1.64·11-s − 0.169·13-s + 1.47·15-s + 2.58·17-s − 3.87·19-s − 21-s + 23-s − 2.83·25-s − 27-s − 0.357·29-s + 3.87·31-s − 1.64·33-s − 1.47·35-s − 4.35·37-s + 0.169·39-s − 9.87·41-s + 5.11·43-s − 1.47·45-s − 8.79·47-s + 49-s − 2.58·51-s + 13.5·53-s − 2.41·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.658·5-s + 0.377·7-s + 0.333·9-s + 0.495·11-s − 0.0469·13-s + 0.380·15-s + 0.627·17-s − 0.888·19-s − 0.218·21-s + 0.208·23-s − 0.566·25-s − 0.192·27-s − 0.0664·29-s + 0.695·31-s − 0.285·33-s − 0.248·35-s − 0.716·37-s + 0.0271·39-s − 1.54·41-s + 0.780·43-s − 0.219·45-s − 1.28·47-s + 0.142·49-s − 0.362·51-s + 1.86·53-s − 0.326·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 + 1.47T + 5T^{2} \) |
| 11 | \( 1 - 1.64T + 11T^{2} \) |
| 13 | \( 1 + 0.169T + 13T^{2} \) |
| 17 | \( 1 - 2.58T + 17T^{2} \) |
| 19 | \( 1 + 3.87T + 19T^{2} \) |
| 29 | \( 1 + 0.357T + 29T^{2} \) |
| 31 | \( 1 - 3.87T + 31T^{2} \) |
| 37 | \( 1 + 4.35T + 37T^{2} \) |
| 41 | \( 1 + 9.87T + 41T^{2} \) |
| 43 | \( 1 - 5.11T + 43T^{2} \) |
| 47 | \( 1 + 8.79T + 47T^{2} \) |
| 53 | \( 1 - 13.5T + 53T^{2} \) |
| 59 | \( 1 - 7.47T + 59T^{2} \) |
| 61 | \( 1 + 3.11T + 61T^{2} \) |
| 67 | \( 1 - 10.1T + 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 + 12.3T + 73T^{2} \) |
| 79 | \( 1 + 0.587T + 79T^{2} \) |
| 83 | \( 1 - 6.10T + 83T^{2} \) |
| 89 | \( 1 + 14.5T + 89T^{2} \) |
| 97 | \( 1 - 4.81T + 97T^{2} \) |
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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
−7.37975355681296725061335438739, −6.91628907930372963836521079968, −6.10599793801371604136394843859, −5.43049236875561190862049851346, −4.64178906631413710725656907329, −4.02230325087071911645390794700, −3.28885341994766267067931154384, −2.11764403899645822654593012790, −1.16066472345007340551399165282, 0,
1.16066472345007340551399165282, 2.11764403899645822654593012790, 3.28885341994766267067931154384, 4.02230325087071911645390794700, 4.64178906631413710725656907329, 5.43049236875561190862049851346, 6.10599793801371604136394843859, 6.91628907930372963836521079968, 7.37975355681296725061335438739