| L(s) = 1 | − 3-s − 7-s + 9-s − 11-s − 13-s + 17-s − 4·19-s + 21-s − 3·23-s − 27-s + 8·29-s + 4·31-s + 33-s + 3·37-s + 39-s − 9·41-s + 8·43-s + 10·47-s − 6·49-s − 51-s − 53-s + 4·57-s + 4·59-s + 11·61-s − 63-s + 4·67-s + 3·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.277·13-s + 0.242·17-s − 0.917·19-s + 0.218·21-s − 0.625·23-s − 0.192·27-s + 1.48·29-s + 0.718·31-s + 0.174·33-s + 0.493·37-s + 0.160·39-s − 1.40·41-s + 1.21·43-s + 1.45·47-s − 6/7·49-s − 0.140·51-s − 0.137·53-s + 0.529·57-s + 0.520·59-s + 1.40·61-s − 0.125·63-s + 0.488·67-s + 0.361·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 15 T + p T^{2} \) |
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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
−14.53779276416492, −13.98953968280858, −13.46170579537291, −12.92890516119586, −12.48566105860226, −11.97717549710942, −11.62591253115127, −10.88962144245779, −10.41225077575555, −10.05864525772371, −9.557684496840087, −8.837050330662739, −8.263444150792172, −7.882944013191339, −7.048783667499702, −6.662598013755061, −6.160117332792315, −5.521244841873675, −5.057042400915140, −4.196981259241655, −4.042353873349116, −2.896744218351639, −2.567384542169322, −1.636453948683892, −0.8099390323260965, 0,
0.8099390323260965, 1.636453948683892, 2.567384542169322, 2.896744218351639, 4.042353873349116, 4.196981259241655, 5.057042400915140, 5.521244841873675, 6.160117332792315, 6.662598013755061, 7.048783667499702, 7.882944013191339, 8.263444150792172, 8.837050330662739, 9.557684496840087, 10.05864525772371, 10.41225077575555, 10.88962144245779, 11.62591253115127, 11.97717549710942, 12.48566105860226, 12.92890516119586, 13.46170579537291, 13.98953968280858, 14.53779276416492
