| L(s) = 1 | + 3-s − 2·9-s + 5·11-s − 7·13-s + 3·17-s + 2·19-s + 8·23-s − 5·27-s − 5·29-s + 10·31-s + 5·33-s − 4·37-s − 7·39-s − 6·41-s + 2·43-s − 7·47-s + 3·51-s + 10·53-s + 2·57-s + 10·59-s − 12·61-s − 2·67-s + 8·69-s + 2·73-s + 7·79-s + 81-s + 4·83-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2/3·9-s + 1.50·11-s − 1.94·13-s + 0.727·17-s + 0.458·19-s + 1.66·23-s − 0.962·27-s − 0.928·29-s + 1.79·31-s + 0.870·33-s − 0.657·37-s − 1.12·39-s − 0.937·41-s + 0.304·43-s − 1.02·47-s + 0.420·51-s + 1.37·53-s + 0.264·57-s + 1.30·59-s − 1.53·61-s − 0.244·67-s + 0.963·69-s + 0.234·73-s + 0.787·79-s + 1/9·81-s + 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.572892730\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.572892730\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−15.36559819066447, −14.94901657750806, −14.74621747117200, −14.02839883989244, −13.77616301487114, −12.95901523880923, −12.28278815081211, −11.79984264356599, −11.53815301338765, −10.66837094095972, −9.811089939594920, −9.633520176387817, −8.935054300716099, −8.485458338906305, −7.710088038907539, −7.145902591185859, −6.701103940186805, −5.831094146406216, −5.132073396841510, −4.637364779915616, −3.688591939455447, −3.115288353280014, −2.511469573650616, −1.616115572634551, −0.6490620925579400,
0.6490620925579400, 1.616115572634551, 2.511469573650616, 3.115288353280014, 3.688591939455447, 4.637364779915616, 5.132073396841510, 5.831094146406216, 6.701103940186805, 7.145902591185859, 7.710088038907539, 8.485458338906305, 8.935054300716099, 9.633520176387817, 9.811089939594920, 10.66837094095972, 11.53815301338765, 11.79984264356599, 12.28278815081211, 12.95901523880923, 13.77616301487114, 14.02839883989244, 14.74621747117200, 14.94901657750806, 15.36559819066447
