| L(s) = 1 | − 3-s − 3·7-s + 9-s + 11-s − 13-s + 17-s + 2·19-s + 3·21-s + 3·23-s − 27-s + 2·29-s − 6·31-s − 33-s + 11·37-s + 39-s − 5·41-s + 4·43-s + 10·47-s + 2·49-s − 51-s + 11·53-s − 2·57-s − 8·59-s − 13·61-s − 3·63-s + 12·67-s − 3·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.13·7-s + 1/3·9-s + 0.301·11-s − 0.277·13-s + 0.242·17-s + 0.458·19-s + 0.654·21-s + 0.625·23-s − 0.192·27-s + 0.371·29-s − 1.07·31-s − 0.174·33-s + 1.80·37-s + 0.160·39-s − 0.780·41-s + 0.609·43-s + 1.45·47-s + 2/7·49-s − 0.140·51-s + 1.51·53-s − 0.264·57-s − 1.04·59-s − 1.66·61-s − 0.377·63-s + 1.46·67-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 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−14.46420705161559, −13.99906381129485, −13.43062042853779, −12.85102570377449, −12.59373106815200, −12.04377775509157, −11.47344404765900, −11.05111619445691, −10.36015201452621, −10.00522172813622, −9.392865951623616, −9.070812542904022, −8.407995395898138, −7.454599821722918, −7.321408562301981, −6.637159945181289, −6.041822082107685, −5.687748759992798, −5.027787130358664, −4.281077514131832, −3.831780802870396, −2.995083989410826, −2.624269688521265, −1.546412882460648, −0.8335028982901347, 0,
0.8335028982901347, 1.546412882460648, 2.624269688521265, 2.995083989410826, 3.831780802870396, 4.281077514131832, 5.027787130358664, 5.687748759992798, 6.041822082107685, 6.637159945181289, 7.321408562301981, 7.454599821722918, 8.407995395898138, 9.070812542904022, 9.392865951623616, 10.00522172813622, 10.36015201452621, 11.05111619445691, 11.47344404765900, 12.04377775509157, 12.59373106815200, 12.85102570377449, 13.43062042853779, 13.99906381129485, 14.46420705161559
