| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s + 2·7-s − 8-s + 9-s + 10-s + 4·11-s + 12-s + 13-s − 2·14-s − 15-s + 16-s + 8·17-s − 18-s − 20-s + 2·21-s − 4·22-s + 6·23-s − 24-s + 25-s − 26-s + 27-s + 2·28-s + 4·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s + 0.288·12-s + 0.277·13-s − 0.534·14-s − 0.258·15-s + 1/4·16-s + 1.94·17-s − 0.235·18-s − 0.223·20-s + 0.436·21-s − 0.852·22-s + 1.25·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s + 0.377·28-s + 0.742·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.634805643\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.634805643\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−13.46264455313928, −12.90679954190904, −12.18331014679868, −11.88037832522698, −11.66713531313155, −10.77138496134122, −10.64871845633071, −9.937899738876794, −9.432610753526800, −8.997824583604285, −8.578054481994376, −8.060858632533863, −7.586368805823985, −7.309524855414605, −6.553900394117755, −6.171577082085859, −5.367707385047700, −4.874644452670104, −4.199427332157511, −3.616962304879712, −3.123587039452195, −2.588847340572221, −1.530086930304891, −1.324202554777468, −0.6804130288416537,
0.6804130288416537, 1.324202554777468, 1.530086930304891, 2.588847340572221, 3.123587039452195, 3.616962304879712, 4.199427332157511, 4.874644452670104, 5.367707385047700, 6.171577082085859, 6.553900394117755, 7.309524855414605, 7.586368805823985, 8.060858632533863, 8.578054481994376, 8.997824583604285, 9.432610753526800, 9.937899738876794, 10.64871845633071, 10.77138496134122, 11.66713531313155, 11.88037832522698, 12.18331014679868, 12.90679954190904, 13.46264455313928
