L(s) = 1 | − 3-s + 7-s − 2·9-s + 3·11-s + 3·13-s + 3·17-s + 2·19-s − 21-s − 23-s + 5·27-s + 3·29-s + 4·31-s − 3·33-s + 8·37-s − 3·39-s − 10·41-s − 4·43-s − 13·47-s + 49-s − 3·51-s − 2·57-s + 8·59-s − 10·61-s − 2·63-s + 6·67-s + 69-s + 6·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.377·7-s − 2/3·9-s + 0.904·11-s + 0.832·13-s + 0.727·17-s + 0.458·19-s − 0.218·21-s − 0.208·23-s + 0.962·27-s + 0.557·29-s + 0.718·31-s − 0.522·33-s + 1.31·37-s − 0.480·39-s − 1.56·41-s − 0.609·43-s − 1.89·47-s + 1/7·49-s − 0.420·51-s − 0.264·57-s + 1.04·59-s − 1.28·61-s − 0.251·63-s + 0.733·67-s + 0.120·69-s + 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−14.52628452165139, −14.00195049950482, −13.52785965983604, −13.02674130586958, −12.23260721496235, −11.87881974973279, −11.52324673127664, −11.08782873863068, −10.53303358539207, −9.782833832300758, −9.605308933323171, −8.608106917275398, −8.415447252086902, −7.910485692428081, −7.090571590438462, −6.472416560511175, −6.193361920109103, −5.529463124573457, −4.993398524990312, −4.454404870918813, −3.633289753004798, −3.214834984590904, −2.434934019731653, −1.410854248109441, −1.065398670791345, 0,
1.065398670791345, 1.410854248109441, 2.434934019731653, 3.214834984590904, 3.633289753004798, 4.454404870918813, 4.993398524990312, 5.529463124573457, 6.193361920109103, 6.472416560511175, 7.090571590438462, 7.910485692428081, 8.415447252086902, 8.608106917275398, 9.605308933323171, 9.782833832300758, 10.53303358539207, 11.08782873863068, 11.52324673127664, 11.87881974973279, 12.23260721496235, 13.02674130586958, 13.52785965983604, 14.00195049950482, 14.52628452165139