| L(s) = 1 | − 2-s + 4-s − 7-s − 8-s − 4·11-s − 3·13-s + 14-s + 16-s + 17-s + 4·22-s + 23-s + 3·26-s − 28-s − 4·31-s − 32-s − 34-s + 11·37-s + 10·41-s − 2·43-s − 4·44-s − 46-s − 11·47-s + 49-s − 3·52-s − 53-s + 56-s + 8·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 1.20·11-s − 0.832·13-s + 0.267·14-s + 1/4·16-s + 0.242·17-s + 0.852·22-s + 0.208·23-s + 0.588·26-s − 0.188·28-s − 0.718·31-s − 0.176·32-s − 0.171·34-s + 1.80·37-s + 1.56·41-s − 0.304·43-s − 0.603·44-s − 0.147·46-s − 1.60·47-s + 1/7·49-s − 0.416·52-s − 0.137·53-s + 0.133·56-s + 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6781469860\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6781469860\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 13 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−14.28103272872260, −13.34440516276360, −13.13326002734500, −12.60596970982153, −12.10892135788932, −11.50217792013025, −10.93184341602323, −10.61802057943176, −9.934631854241442, −9.541572114189493, −9.246787773676413, −8.338304507942398, −8.004137843851223, −7.499390427276532, −7.043855863299145, −6.375049733582407, −5.785414389617559, −5.272513393934164, −4.644511592852209, −3.963722924803932, −3.071103867813711, −2.689026196406530, −2.091523378987585, −1.187044666863930, −0.3210623265369088,
0.3210623265369088, 1.187044666863930, 2.091523378987585, 2.689026196406530, 3.071103867813711, 3.963722924803932, 4.644511592852209, 5.272513393934164, 5.785414389617559, 6.375049733582407, 7.043855863299145, 7.499390427276532, 8.004137843851223, 8.338304507942398, 9.246787773676413, 9.541572114189493, 9.934631854241442, 10.61802057943176, 10.93184341602323, 11.50217792013025, 12.10892135788932, 12.60596970982153, 13.13326002734500, 13.34440516276360, 14.28103272872260
