| L(s) = 1 | − 4·7-s + 11-s − 4·19-s − 5·25-s − 10·31-s − 2·37-s − 6·41-s + 10·43-s + 9·49-s − 6·53-s + 2·61-s + 2·67-s − 12·71-s + 10·73-s − 4·77-s + 10·79-s + 12·83-s + 12·89-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 0.301·11-s − 0.917·19-s − 25-s − 1.79·31-s − 0.328·37-s − 0.937·41-s + 1.52·43-s + 9/7·49-s − 0.824·53-s + 0.256·61-s + 0.244·67-s − 1.42·71-s + 1.17·73-s − 0.455·77-s + 1.12·79-s + 1.31·83-s + 1.27·89-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 267696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 267696 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5741567842\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5741567842\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−12.82820887212169, −12.32111093369997, −12.07588866843805, −11.34562656105888, −10.94941252078181, −10.38518499496911, −10.08896500188449, −9.445763533676192, −9.107251801410602, −8.874336010725693, −7.998572712677615, −7.685181597583018, −7.015005807099671, −6.659519672423863, −6.152671417509208, −5.800347044326772, −5.215480205651215, −4.524708091804543, −3.947406459597467, −3.473596367232296, −3.193229720664006, −2.157957669554428, −2.072900196774872, −1.021924612559432, −0.2223189999380378,
0.2223189999380378, 1.021924612559432, 2.072900196774872, 2.157957669554428, 3.193229720664006, 3.473596367232296, 3.947406459597467, 4.524708091804543, 5.215480205651215, 5.800347044326772, 6.152671417509208, 6.659519672423863, 7.015005807099671, 7.685181597583018, 7.998572712677615, 8.874336010725693, 9.107251801410602, 9.445763533676192, 10.08896500188449, 10.38518499496911, 10.94941252078181, 11.34562656105888, 12.07588866843805, 12.32111093369997, 12.82820887212169
