| L(s) = 1 | + 2·5-s − 4·11-s − 2·13-s − 2·17-s − 4·19-s + 8·23-s − 25-s − 6·29-s + 8·31-s + 6·37-s + 6·41-s + 4·43-s − 7·49-s + 2·53-s − 8·55-s − 4·59-s − 2·61-s − 4·65-s − 4·67-s − 8·71-s + 10·73-s − 8·79-s + 4·83-s − 4·85-s + 6·89-s − 8·95-s + 2·97-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.20·11-s − 0.554·13-s − 0.485·17-s − 0.917·19-s + 1.66·23-s − 1/5·25-s − 1.11·29-s + 1.43·31-s + 0.986·37-s + 0.937·41-s + 0.609·43-s − 49-s + 0.274·53-s − 1.07·55-s − 0.520·59-s − 0.256·61-s − 0.496·65-s − 0.488·67-s − 0.949·71-s + 1.17·73-s − 0.900·79-s + 0.439·83-s − 0.433·85-s + 0.635·89-s − 0.820·95-s + 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9732684211\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9732684211\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−14.66535545665708588625458363833, −13.38110032732463692857243615568, −12.79661702509897009228746961274, −11.17432181718975973992131540327, −10.16546244400809085298891139382, −9.065170023749915638598404079220, −7.62403207119791885504595168937, −6.14969218484336320739052245813, −4.83326855247570334999188527992, −2.51659494204614894722398853037,
2.51659494204614894722398853037, 4.83326855247570334999188527992, 6.14969218484336320739052245813, 7.62403207119791885504595168937, 9.065170023749915638598404079220, 10.16546244400809085298891139382, 11.17432181718975973992131540327, 12.79661702509897009228746961274, 13.38110032732463692857243615568, 14.66535545665708588625458363833
