L(s) = 1 | + 5-s + 2·7-s + 3·11-s + 2·13-s + 3·17-s − 6·19-s + 23-s + 25-s + 29-s − 8·31-s + 2·35-s − 9·37-s − 9·43-s − 3·49-s + 14·53-s + 3·55-s + 5·59-s − 2·61-s + 2·65-s − 3·67-s − 2·71-s + 10·73-s + 6·77-s − 11·79-s − 6·83-s + 3·85-s − 6·89-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.755·7-s + 0.904·11-s + 0.554·13-s + 0.727·17-s − 1.37·19-s + 0.208·23-s + 1/5·25-s + 0.185·29-s − 1.43·31-s + 0.338·35-s − 1.47·37-s − 1.37·43-s − 3/7·49-s + 1.92·53-s + 0.404·55-s + 0.650·59-s − 0.256·61-s + 0.248·65-s − 0.366·67-s − 0.237·71-s + 1.17·73-s + 0.683·77-s − 1.23·79-s − 0.658·83-s + 0.325·85-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−14.39458939898147, −13.78218411521221, −13.31941571417402, −12.84010764542329, −12.16858707998790, −11.90166402487476, −11.14266308591622, −10.89709482188350, −10.25158826542317, −9.837877297890621, −9.119506141508758, −8.607192354803251, −8.421292080687044, −7.660872799227423, −6.843537972649458, −6.777430486541587, −5.873579543890531, −5.494723774126846, −4.908062724028573, −4.182894657739946, −3.733421947063766, −3.094576404006552, −2.142362169010437, −1.677896394992092, −1.117391658102387, 0,
1.117391658102387, 1.677896394992092, 2.142362169010437, 3.094576404006552, 3.733421947063766, 4.182894657739946, 4.908062724028573, 5.494723774126846, 5.873579543890531, 6.777430486541587, 6.843537972649458, 7.660872799227423, 8.421292080687044, 8.607192354803251, 9.119506141508758, 9.837877297890621, 10.25158826542317, 10.89709482188350, 11.14266308591622, 11.90166402487476, 12.16858707998790, 12.84010764542329, 13.31941571417402, 13.78218411521221, 14.39458939898147