| L(s) = 1 | − 2·5-s − 7-s − 4·11-s − 2·17-s + 4·19-s − 25-s + 2·29-s + 2·35-s + 2·37-s − 6·41-s + 4·43-s − 8·47-s + 49-s + 10·53-s + 8·55-s − 4·59-s − 2·61-s + 4·67-s − 2·73-s + 4·77-s + 8·79-s − 12·83-s + 4·85-s − 6·89-s − 8·95-s − 10·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.377·7-s − 1.20·11-s − 0.485·17-s + 0.917·19-s − 1/5·25-s + 0.371·29-s + 0.338·35-s + 0.328·37-s − 0.937·41-s + 0.609·43-s − 1.16·47-s + 1/7·49-s + 1.37·53-s + 1.07·55-s − 0.520·59-s − 0.256·61-s + 0.488·67-s − 0.234·73-s + 0.455·77-s + 0.900·79-s − 1.31·83-s + 0.433·85-s − 0.635·89-s − 0.820·95-s − 1.01·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−13.96869957387071, −13.72464846730589, −13.16560189889684, −12.66788067060960, −12.23545215038233, −11.62347400861667, −11.30956118252231, −10.73318271701178, −10.14611882806703, −9.804194189360675, −9.133172363098379, −8.552449144419660, −8.006593504632034, −7.707062078404043, −7.049148573777391, −6.670328675447847, −5.821070184940284, −5.411302174149522, −4.761195586010177, −4.231679959061066, −3.575971833301466, −3.017939130899475, −2.502441249839436, −1.656513888199206, −0.6820319257978303, 0,
0.6820319257978303, 1.656513888199206, 2.502441249839436, 3.017939130899475, 3.575971833301466, 4.231679959061066, 4.761195586010177, 5.411302174149522, 5.821070184940284, 6.670328675447847, 7.049148573777391, 7.707062078404043, 8.006593504632034, 8.552449144419660, 9.133172363098379, 9.804194189360675, 10.14611882806703, 10.73318271701178, 11.30956118252231, 11.62347400861667, 12.23545215038233, 12.66788067060960, 13.16560189889684, 13.72464846730589, 13.96869957387071
