L(s) = 1 | − 3-s + 3.73·5-s − 7-s + 9-s + 4.20·11-s + 0.470·13-s − 3.73·15-s − 4.20·17-s − 4.20·19-s + 21-s − 23-s + 8.93·25-s − 27-s − 0.651·29-s − 6.20·31-s − 4.20·33-s − 3.73·35-s − 5.75·37-s − 0.470·39-s − 3.75·41-s − 11.9·43-s + 3.73·45-s − 6.61·47-s + 49-s + 4.20·51-s − 3.28·53-s + 15.6·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.66·5-s − 0.377·7-s + 0.333·9-s + 1.26·11-s + 0.130·13-s − 0.963·15-s − 1.01·17-s − 0.964·19-s + 0.218·21-s − 0.208·23-s + 1.78·25-s − 0.192·27-s − 0.120·29-s − 1.11·31-s − 0.731·33-s − 0.631·35-s − 0.946·37-s − 0.0753·39-s − 0.586·41-s − 1.82·43-s + 0.556·45-s − 0.964·47-s + 0.142·49-s + 0.588·51-s − 0.451·53-s + 2.11·55-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 | \( 1 - 3.73T + 5T^{2} \) |
| 11 | \( 1 - 4.20T + 11T^{2} \) |
| 13 | \( 1 - 0.470T + 13T^{2} \) |
| 17 | \( 1 + 4.20T + 17T^{2} \) |
| 19 | \( 1 + 4.20T + 19T^{2} \) |
| 29 | \( 1 + 0.651T + 29T^{2} \) |
| 31 | \( 1 + 6.20T + 31T^{2} \) |
| 37 | \( 1 + 5.75T + 37T^{2} \) |
| 41 | \( 1 + 3.75T + 41T^{2} \) |
| 43 | \( 1 + 11.9T + 43T^{2} \) |
| 47 | \( 1 + 6.61T + 47T^{2} \) |
| 53 | \( 1 + 3.28T + 53T^{2} \) |
| 59 | \( 1 + 8.58T + 59T^{2} \) |
| 61 | \( 1 - 9.08T + 61T^{2} \) |
| 67 | \( 1 + 5.81T + 67T^{2} \) |
| 71 | \( 1 + 9.85T + 71T^{2} \) |
| 73 | \( 1 - 3.84T + 73T^{2} \) |
| 79 | \( 1 + 15.5T + 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 + 4.11T + 97T^{2} \) |
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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
−7.12827401633113394606704621576, −6.58692182326966959257123667821, −6.25877555838090185319195377519, −5.55554402951931532157125769617, −4.82310104037141074413994064301, −4.01611821949517285197623711853, −3.06956945956671179866515754020, −1.87018092908641218490858084377, −1.58920174938450651156538757796, 0,
1.58920174938450651156538757796, 1.87018092908641218490858084377, 3.06956945956671179866515754020, 4.01611821949517285197623711853, 4.82310104037141074413994064301, 5.55554402951931532157125769617, 6.25877555838090185319195377519, 6.58692182326966959257123667821, 7.12827401633113394606704621576