| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s − 3·7-s + 8-s − 2·9-s − 10-s + 2·11-s − 12-s + 3·13-s − 3·14-s + 15-s + 16-s − 3·17-s − 2·18-s + 3·19-s − 20-s + 3·21-s + 2·22-s − 4·23-s − 24-s + 25-s + 3·26-s + 5·27-s − 3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 1.13·7-s + 0.353·8-s − 2/3·9-s − 0.316·10-s + 0.603·11-s − 0.288·12-s + 0.832·13-s − 0.801·14-s + 0.258·15-s + 1/4·16-s − 0.727·17-s − 0.471·18-s + 0.688·19-s − 0.223·20-s + 0.654·21-s + 0.426·22-s − 0.834·23-s − 0.204·24-s + 1/5·25-s + 0.588·26-s + 0.962·27-s − 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.994816281\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.994816281\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| 139 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−14.59761678505409, −14.28585112333023, −13.65411975029741, −13.28527487344348, −12.50131593243492, −12.35079517599279, −11.57963615631056, −11.28256208712500, −10.85605940085805, −10.09455082859168, −9.533786433484642, −8.991863358851340, −8.315355482114203, −7.748413283706885, −6.984594114944669, −6.438935900538640, −6.041809313883596, −5.697820828730088, −4.770993771465571, −4.167793415794051, −3.716961124315996, −2.920694884931475, −2.529000142094806, −1.281202001384403, −0.5067334838682477,
0.5067334838682477, 1.281202001384403, 2.529000142094806, 2.920694884931475, 3.716961124315996, 4.167793415794051, 4.770993771465571, 5.697820828730088, 6.041809313883596, 6.438935900538640, 6.984594114944669, 7.748413283706885, 8.315355482114203, 8.991863358851340, 9.533786433484642, 10.09455082859168, 10.85605940085805, 11.28256208712500, 11.57963615631056, 12.35079517599279, 12.50131593243492, 13.28527487344348, 13.65411975029741, 14.28585112333023, 14.59761678505409
