L(s) = 1 | − 2·4-s + 7-s + 11-s + 13-s + 4·16-s − 17-s − 4·19-s − 3·23-s − 2·28-s + 8·29-s − 4·31-s − 3·37-s + 9·41-s + 8·43-s − 2·44-s + 10·47-s − 6·49-s − 2·52-s − 53-s − 4·59-s − 11·61-s − 8·64-s + 4·67-s + 2·68-s + 71-s − 14·73-s + 8·76-s + ⋯ |
L(s) = 1 | − 4-s + 0.377·7-s + 0.301·11-s + 0.277·13-s + 16-s − 0.242·17-s − 0.917·19-s − 0.625·23-s − 0.377·28-s + 1.48·29-s − 0.718·31-s − 0.493·37-s + 1.40·41-s + 1.21·43-s − 0.301·44-s + 1.45·47-s − 6/7·49-s − 0.277·52-s − 0.137·53-s − 0.520·59-s − 1.40·61-s − 64-s + 0.488·67-s + 0.242·68-s + 0.118·71-s − 1.63·73-s + 0.917·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.371039565\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.371039565\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 2 | \( 1 + p T^{2} \) |
| 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
−8.897663172163547484325799351713, −8.089771937793132184499294415351, −7.43759897448906982672581790235, −6.30669141510903739665096402460, −5.73278414244578144806247587705, −4.62157204419073793031116621923, −4.26067299548702583887435194086, −3.24149746825360535310979514555, −1.99562059501787958433191990418, −0.73375372520195490235059592257,
0.73375372520195490235059592257, 1.99562059501787958433191990418, 3.24149746825360535310979514555, 4.26067299548702583887435194086, 4.62157204419073793031116621923, 5.73278414244578144806247587705, 6.30669141510903739665096402460, 7.43759897448906982672581790235, 8.089771937793132184499294415351, 8.897663172163547484325799351713