| L(s) = 1 | − 3·5-s + 7-s + 13-s + 5·19-s + 6·23-s + 4·25-s − 3·29-s + 4·31-s − 3·35-s − 7·37-s − 12·41-s + 2·43-s + 3·47-s + 49-s − 6·53-s + 3·59-s − 2·61-s − 3·65-s + 67-s + 12·71-s + 7·73-s − 4·79-s − 6·89-s + 91-s − 15·95-s + 2·97-s + 101-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 0.377·7-s + 0.277·13-s + 1.14·19-s + 1.25·23-s + 4/5·25-s − 0.557·29-s + 0.718·31-s − 0.507·35-s − 1.15·37-s − 1.87·41-s + 0.304·43-s + 0.437·47-s + 1/7·49-s − 0.824·53-s + 0.390·59-s − 0.256·61-s − 0.372·65-s + 0.122·67-s + 1.42·71-s + 0.819·73-s − 0.450·79-s − 0.635·89-s + 0.104·91-s − 1.53·95-s + 0.203·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 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
−13.92158514064213, −13.32410242711670, −12.68594697045056, −12.32104767518090, −11.75296624571169, −11.37145389202316, −11.15113583665975, −10.39411927909764, −10.03844973044695, −9.219263182523434, −8.903566807912932, −8.257259975768488, −7.929082105497004, −7.390245443968276, −6.929424302385797, −6.493752562361380, −5.585026745110621, −5.123717184746832, −4.707692262499895, −4.012460911274149, −3.418037020937437, −3.182097229771616, −2.279179526684246, −1.435981361573723, −0.8329392822674146, 0,
0.8329392822674146, 1.435981361573723, 2.279179526684246, 3.182097229771616, 3.418037020937437, 4.012460911274149, 4.707692262499895, 5.123717184746832, 5.585026745110621, 6.493752562361380, 6.929424302385797, 7.390245443968276, 7.929082105497004, 8.257259975768488, 8.903566807912932, 9.219263182523434, 10.03844973044695, 10.39411927909764, 11.15113583665975, 11.37145389202316, 11.75296624571169, 12.32104767518090, 12.68594697045056, 13.32410242711670, 13.92158514064213
