| L(s) = 1 | + 2·5-s + 2·7-s + 4·11-s − 13-s − 2·19-s + 4·23-s − 25-s + 2·31-s + 4·35-s − 10·37-s + 2·41-s + 8·43-s − 3·49-s + 12·53-s + 8·55-s + 12·59-s − 6·61-s − 2·65-s − 6·67-s − 8·71-s − 2·73-s + 8·77-s + 12·79-s − 4·83-s + 14·89-s − 2·91-s − 4·95-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.755·7-s + 1.20·11-s − 0.277·13-s − 0.458·19-s + 0.834·23-s − 1/5·25-s + 0.359·31-s + 0.676·35-s − 1.64·37-s + 0.312·41-s + 1.21·43-s − 3/7·49-s + 1.64·53-s + 1.07·55-s + 1.56·59-s − 0.768·61-s − 0.248·65-s − 0.733·67-s − 0.949·71-s − 0.234·73-s + 0.911·77-s + 1.35·79-s − 0.439·83-s + 1.48·89-s − 0.209·91-s − 0.410·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.114780621\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.114780621\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−10.04843529709332670174676581445, −9.153667018132570450903651076940, −8.612568759993176324608702875332, −7.44699838933626723226640796085, −6.61084334022363680034133660955, −5.73335318755003699286776095361, −4.82890466759038882574961362490, −3.80548435074655755032944611187, −2.36222722805652747794129158747, −1.32780807021178389715142739482,
1.32780807021178389715142739482, 2.36222722805652747794129158747, 3.80548435074655755032944611187, 4.82890466759038882574961362490, 5.73335318755003699286776095361, 6.61084334022363680034133660955, 7.44699838933626723226640796085, 8.612568759993176324608702875332, 9.153667018132570450903651076940, 10.04843529709332670174676581445
