L(s) = 1 | − 3-s + 9-s − 4·11-s − 6·13-s − 6·17-s + 4·19-s + 8·23-s − 27-s − 10·29-s − 4·31-s + 4·33-s − 6·37-s + 6·39-s − 6·41-s + 4·43-s − 12·47-s + 6·51-s + 6·53-s − 4·57-s + 4·59-s − 2·61-s + 4·67-s − 8·69-s − 2·73-s − 8·79-s + 81-s + 12·83-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.20·11-s − 1.66·13-s − 1.45·17-s + 0.917·19-s + 1.66·23-s − 0.192·27-s − 1.85·29-s − 0.718·31-s + 0.696·33-s − 0.986·37-s + 0.960·39-s − 0.937·41-s + 0.609·43-s − 1.75·47-s + 0.840·51-s + 0.824·53-s − 0.529·57-s + 0.520·59-s − 0.256·61-s + 0.488·67-s − 0.963·69-s − 0.234·73-s − 0.900·79-s + 1/9·81-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 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 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−13.07970197284343, −12.71514678741292, −12.30850320548454, −11.61018452085202, −11.24796952957967, −10.98699189154425, −10.32220842269963, −9.967871312250315, −9.455833048501540, −8.991046074587451, −8.537465985114107, −7.715034832570615, −7.432643919935315, −6.989992842065457, −6.667343139244253, −5.785490845190386, −5.249500593804753, −5.092550087909725, −4.639686386675219, −3.873490925392040, −3.226379029448415, −2.680491651464082, −2.097090613706049, −1.576851849451683, −0.5020548711091982, 0,
0.5020548711091982, 1.576851849451683, 2.097090613706049, 2.680491651464082, 3.226379029448415, 3.873490925392040, 4.639686386675219, 5.092550087909725, 5.249500593804753, 5.785490845190386, 6.667343139244253, 6.989992842065457, 7.432643919935315, 7.715034832570615, 8.537465985114107, 8.991046074587451, 9.455833048501540, 9.967871312250315, 10.32220842269963, 10.98699189154425, 11.24796952957967, 11.61018452085202, 12.30850320548454, 12.71514678741292, 13.07970197284343