| L(s) = 1 | + 5-s − 4·11-s − 2·13-s − 6·17-s + 8·23-s + 25-s + 10·29-s + 8·31-s − 2·37-s − 2·41-s − 8·43-s + 4·47-s + 10·53-s − 4·55-s − 4·59-s − 6·61-s − 2·65-s + 12·71-s + 6·73-s − 8·79-s + 4·83-s − 6·85-s + 14·89-s − 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.20·11-s − 0.554·13-s − 1.45·17-s + 1.66·23-s + 1/5·25-s + 1.85·29-s + 1.43·31-s − 0.328·37-s − 0.312·41-s − 1.21·43-s + 0.583·47-s + 1.37·53-s − 0.539·55-s − 0.520·59-s − 0.768·61-s − 0.248·65-s + 1.42·71-s + 0.702·73-s − 0.900·79-s + 0.439·83-s − 0.650·85-s + 1.48·89-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.212464403\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.212464403\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.42401208524928, −13.03276344149072, −12.48283983505357, −12.02539740584059, −11.46672435090554, −10.92925090154347, −10.42484610877510, −10.18010539250763, −9.607519975674521, −8.943011195068285, −8.554831156599074, −8.203337076449303, −7.360528537729318, −7.086539374366253, −6.366204785436065, −6.178855702742294, −5.112056361193689, −4.938710303931401, −4.618519507813586, −3.680844620636861, −2.929646765156930, −2.564011170994487, −2.095495286164341, −1.125021941313194, −0.4742658527660992,
0.4742658527660992, 1.125021941313194, 2.095495286164341, 2.564011170994487, 2.929646765156930, 3.680844620636861, 4.618519507813586, 4.938710303931401, 5.112056361193689, 6.178855702742294, 6.366204785436065, 7.086539374366253, 7.360528537729318, 8.203337076449303, 8.554831156599074, 8.943011195068285, 9.607519975674521, 10.18010539250763, 10.42484610877510, 10.92925090154347, 11.46672435090554, 12.02539740584059, 12.48283983505357, 13.03276344149072, 13.42401208524928
