L(s) = 1 | + (0.450 + 1.38i)3-s + (0.309 − 0.951i)4-s + (−0.866 − 0.629i)5-s + (0.0388 − 0.119i)7-s + (−0.910 + 0.661i)9-s + (0.876 − 0.481i)11-s + 1.45·12-s + (0.688 − 0.500i)13-s + (0.482 − 1.48i)15-s + (−0.809 − 0.587i)16-s + (−0.866 + 0.629i)20-s + 0.183·21-s + (0.0458 + 0.141i)25-s + (−0.148 − 0.107i)27-s + (−0.101 − 0.0738i)28-s + ⋯ |
L(s) = 1 | + (0.450 + 1.38i)3-s + (0.309 − 0.951i)4-s + (−0.866 − 0.629i)5-s + (0.0388 − 0.119i)7-s + (−0.910 + 0.661i)9-s + (0.876 − 0.481i)11-s + 1.45·12-s + (0.688 − 0.500i)13-s + (0.482 − 1.48i)15-s + (−0.809 − 0.587i)16-s + (−0.866 + 0.629i)20-s + 0.183·21-s + (0.0458 + 0.141i)25-s + (−0.148 − 0.107i)27-s + (−0.101 − 0.0738i)28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.236574814\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.236574814\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (-0.876 + 0.481i)T \) |
| 131 | \( 1 - T \) |
good | 2 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 3 | \( 1 + (-0.450 - 1.38i)T + (-0.809 + 0.587i)T^{2} \) |
| 5 | \( 1 + (0.866 + 0.629i)T + (0.309 + 0.951i)T^{2} \) |
| 7 | \( 1 + (-0.0388 + 0.119i)T + (-0.809 - 0.587i)T^{2} \) |
| 13 | \( 1 + (-0.688 + 0.500i)T + (0.309 - 0.951i)T^{2} \) |
| 17 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 19 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 31 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (0.115 + 0.356i)T + (-0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 - 1.93T + T^{2} \) |
| 47 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (0.574 - 1.76i)T + (-0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (0.866 + 0.629i)T + (0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 89 | \( 1 + 1.61T + T^{2} \) |
| 97 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
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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
−9.580259923864516423022667016444, −9.048984956995513483200909595013, −8.417226463755459159253266039607, −7.40438601401775813208167800281, −6.20029256680256205240368208765, −5.44910696305104589864071686409, −4.40726590638176435793174447201, −3.97786415092820127181403263400, −2.88046755074379016906650741098, −1.11721299176090846084419932670,
1.59425175736476051072354290947, 2.62868416116225390004350528640, 3.55678977908372330755360759176, 4.31923123351299490106231043210, 6.11204283772777489031555445282, 6.86109725515458779830432669803, 7.27803117905320219809858663252, 7.975835542717261791058243860848, 8.611541943485471284743059171884, 9.437090611335515177866054918893