| L(s) = 1 | + 2.73·5-s + 7-s + 4·11-s + 5.46·13-s + 6.73·17-s + 19-s − 6.92·23-s + 2.46·25-s + 6.73·29-s − 2·31-s + 2.73·35-s + 8.92·37-s − 4.92·41-s − 8.92·43-s + 10.1·47-s + 49-s + 9.66·53-s + 10.9·55-s − 2.92·59-s − 12.9·61-s + 14.9·65-s − 1.46·67-s + 6.19·71-s + 10.3·73-s + 4·77-s − 10.9·79-s + 2.19·83-s + ⋯ |
| L(s) = 1 | + 1.22·5-s + 0.377·7-s + 1.20·11-s + 1.51·13-s + 1.63·17-s + 0.229·19-s − 1.44·23-s + 0.492·25-s + 1.25·29-s − 0.359·31-s + 0.461·35-s + 1.46·37-s − 0.769·41-s − 1.36·43-s + 1.48·47-s + 0.142·49-s + 1.32·53-s + 1.47·55-s − 0.381·59-s − 1.65·61-s + 1.85·65-s − 0.178·67-s + 0.735·71-s + 1.21·73-s + 0.455·77-s − 1.22·79-s + 0.241·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.893801649\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.893801649\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 2.73T + 5T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 - 5.46T + 13T^{2} \) |
| 17 | \( 1 - 6.73T + 17T^{2} \) |
| 23 | \( 1 + 6.92T + 23T^{2} \) |
| 29 | \( 1 - 6.73T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 8.92T + 37T^{2} \) |
| 41 | \( 1 + 4.92T + 41T^{2} \) |
| 43 | \( 1 + 8.92T + 43T^{2} \) |
| 47 | \( 1 - 10.1T + 47T^{2} \) |
| 53 | \( 1 - 9.66T + 53T^{2} \) |
| 59 | \( 1 + 2.92T + 59T^{2} \) |
| 61 | \( 1 + 12.9T + 61T^{2} \) |
| 67 | \( 1 + 1.46T + 67T^{2} \) |
| 71 | \( 1 - 6.19T + 71T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 - 2.19T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 15.8T + 97T^{2} \) |
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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
−7.82101754553764157746297941289, −6.82766382421861412503423129086, −6.08999287243871066865960338651, −5.90501916147573538269381378539, −5.07992508981003439914161783340, −4.06271904862023123956389879638, −3.54088591409844500854624800673, −2.51367681212303007485765341692, −1.49063448981941371951529877006, −1.12594340283790499975146702965,
1.12594340283790499975146702965, 1.49063448981941371951529877006, 2.51367681212303007485765341692, 3.54088591409844500854624800673, 4.06271904862023123956389879638, 5.07992508981003439914161783340, 5.90501916147573538269381378539, 6.08999287243871066865960338651, 6.82766382421861412503423129086, 7.82101754553764157746297941289
