L(s) = 1 | + 5-s + 7-s + 4·11-s + 4·13-s + 4·19-s + 23-s + 25-s − 2·29-s − 4·31-s + 35-s − 2·37-s − 12·41-s − 4·43-s − 8·47-s + 49-s − 10·53-s + 4·55-s + 6·59-s − 12·61-s + 4·65-s − 4·67-s − 12·71-s + 10·73-s + 4·77-s − 2·79-s + 14·83-s + 6·89-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 0.377·7-s + 1.20·11-s + 1.10·13-s + 0.917·19-s + 0.208·23-s + 1/5·25-s − 0.371·29-s − 0.718·31-s + 0.169·35-s − 0.328·37-s − 1.87·41-s − 0.609·43-s − 1.16·47-s + 1/7·49-s − 1.37·53-s + 0.539·55-s + 0.781·59-s − 1.53·61-s + 0.496·65-s − 0.488·67-s − 1.42·71-s + 1.17·73-s + 0.455·77-s − 0.225·79-s + 1.53·83-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.87433022901236, −13.37590904618917, −13.03662961259033, −12.31623091730781, −11.81007974544239, −11.47345173252551, −11.01310711643562, −10.42863608734072, −9.946582328125381, −9.313604269069554, −9.013021042040114, −8.528444526000400, −7.916810405552838, −7.410533938712576, −6.662744984664297, −6.438087667802850, −5.833721007840751, −5.174824537280993, −4.810413266276228, −4.016134257930991, −3.407337419806602, −3.169804466442259, −2.027355324234249, −1.526830096355585, −1.139986498242309, 0,
1.139986498242309, 1.526830096355585, 2.027355324234249, 3.169804466442259, 3.407337419806602, 4.016134257930991, 4.810413266276228, 5.174824537280993, 5.833721007840751, 6.438087667802850, 6.662744984664297, 7.410533938712576, 7.916810405552838, 8.528444526000400, 9.013021042040114, 9.313604269069554, 9.946582328125381, 10.42863608734072, 11.01310711643562, 11.47345173252551, 11.81007974544239, 12.31623091730781, 13.03662961259033, 13.37590904618917, 13.87433022901236