L(s) = 1 | + 3-s − 7-s + 9-s + 11-s − 5·13-s + 3·17-s + 3·19-s − 21-s − 4·23-s + 27-s + 10·31-s + 33-s + 37-s − 5·39-s − 7·41-s + 10·43-s − 10·47-s + 49-s + 3·51-s + 53-s + 3·57-s + 6·59-s − 61-s − 63-s − 5·67-s − 4·69-s − 9·71-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.301·11-s − 1.38·13-s + 0.727·17-s + 0.688·19-s − 0.218·21-s − 0.834·23-s + 0.192·27-s + 1.79·31-s + 0.174·33-s + 0.164·37-s − 0.800·39-s − 1.09·41-s + 1.52·43-s − 1.45·47-s + 1/7·49-s + 0.420·51-s + 0.137·53-s + 0.397·57-s + 0.781·59-s − 0.128·61-s − 0.125·63-s − 0.610·67-s − 0.481·69-s − 1.06·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.597532839\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.597532839\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−12.44585034934942, −12.02667298315235, −11.78008474258137, −11.27929614766359, −10.42414588598834, −10.12077512904793, −9.829455134941953, −9.425210904210859, −8.898416441107478, −8.364137781223687, −7.865549422556854, −7.531259855106902, −7.099299732716836, −6.382363838823954, −6.230667512406352, −5.262050807919402, −5.155433359446327, −4.336066296127704, −4.037960063712602, −3.289185304195719, −2.884966272373617, −2.453312748184026, −1.765075552548590, −1.131491372261151, −0.4152647462078905,
0.4152647462078905, 1.131491372261151, 1.765075552548590, 2.453312748184026, 2.884966272373617, 3.289185304195719, 4.037960063712602, 4.336066296127704, 5.155433359446327, 5.262050807919402, 6.230667512406352, 6.382363838823954, 7.099299732716836, 7.531259855106902, 7.865549422556854, 8.364137781223687, 8.898416441107478, 9.425210904210859, 9.829455134941953, 10.12077512904793, 10.42414588598834, 11.27929614766359, 11.78008474258137, 12.02667298315235, 12.44585034934942