| L(s) = 1 | − 4·7-s + 4·11-s + 13-s + 2·17-s − 4·19-s + 2·23-s + 2·29-s + 6·31-s − 2·37-s + 2·41-s − 4·43-s + 12·47-s + 9·49-s + 8·59-s + 6·61-s + 10·67-s − 8·71-s − 4·73-s − 16·77-s + 4·79-s + 4·83-s − 2·89-s − 4·91-s + 12·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 1.20·11-s + 0.277·13-s + 0.485·17-s − 0.917·19-s + 0.417·23-s + 0.371·29-s + 1.07·31-s − 0.328·37-s + 0.312·41-s − 0.609·43-s + 1.75·47-s + 9/7·49-s + 1.04·59-s + 0.768·61-s + 1.22·67-s − 0.949·71-s − 0.468·73-s − 1.82·77-s + 0.450·79-s + 0.439·83-s − 0.211·89-s − 0.419·91-s + 1.21·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.322959846\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.322959846\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−13.84158091859518, −13.24396716251121, −12.85340348108570, −12.42893246785684, −11.82267473503397, −11.57959475839683, −10.72496053413686, −10.30718448381779, −9.872816645287855, −9.365338472325123, −8.809689337464588, −8.554371782886848, −7.739593042671581, −7.067258951373742, −6.650813051974930, −6.270386159966148, −5.810108919220639, −5.072091132929416, −4.302303943246503, −3.836171253614418, −3.357433463912766, −2.713912519999588, −2.065172150129421, −1.088038094880112, −0.5571951509972343,
0.5571951509972343, 1.088038094880112, 2.065172150129421, 2.713912519999588, 3.357433463912766, 3.836171253614418, 4.302303943246503, 5.072091132929416, 5.810108919220639, 6.270386159966148, 6.650813051974930, 7.067258951373742, 7.739593042671581, 8.554371782886848, 8.809689337464588, 9.365338472325123, 9.872816645287855, 10.30718448381779, 10.72496053413686, 11.57959475839683, 11.82267473503397, 12.42893246785684, 12.85340348108570, 13.24396716251121, 13.84158091859518
