L(s) = 1 | + 1.48·2-s + 3-s + 0.193·4-s − 4.15·5-s + 1.48·6-s + 7-s − 2.67·8-s + 9-s − 6.15·10-s + 3.19·11-s + 0.193·12-s + 1.48·14-s − 4.15·15-s − 4.35·16-s − 3.35·17-s + 1.48·18-s + 2.38·19-s − 0.806·20-s + 21-s + 4.73·22-s − 0.387·23-s − 2.67·24-s + 12.2·25-s + 27-s + 0.193·28-s − 7.92·29-s − 6.15·30-s + ⋯ |
L(s) = 1 | + 1.04·2-s + 0.577·3-s + 0.0969·4-s − 1.85·5-s + 0.604·6-s + 0.377·7-s − 0.945·8-s + 0.333·9-s − 1.94·10-s + 0.963·11-s + 0.0559·12-s + 0.395·14-s − 1.07·15-s − 1.08·16-s − 0.812·17-s + 0.349·18-s + 0.547·19-s − 0.180·20-s + 0.218·21-s + 1.00·22-s − 0.0808·23-s − 0.546·24-s + 2.45·25-s + 0.192·27-s + 0.0366·28-s − 1.47·29-s − 1.12·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.440125219\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.440125219\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 1.48T + 2T^{2} \) |
| 5 | \( 1 + 4.15T + 5T^{2} \) |
| 11 | \( 1 - 3.19T + 11T^{2} \) |
| 17 | \( 1 + 3.35T + 17T^{2} \) |
| 19 | \( 1 - 2.38T + 19T^{2} \) |
| 23 | \( 1 + 0.387T + 23T^{2} \) |
| 29 | \( 1 + 7.92T + 29T^{2} \) |
| 31 | \( 1 - 10.7T + 31T^{2} \) |
| 37 | \( 1 - 1.61T + 37T^{2} \) |
| 41 | \( 1 - 1.45T + 41T^{2} \) |
| 43 | \( 1 + 1.92T + 43T^{2} \) |
| 47 | \( 1 - 3.76T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 6.15T + 59T^{2} \) |
| 61 | \( 1 - 14.4T + 61T^{2} \) |
| 67 | \( 1 - 5.61T + 67T^{2} \) |
| 71 | \( 1 - 11.8T + 71T^{2} \) |
| 73 | \( 1 - 15.6T + 73T^{2} \) |
| 79 | \( 1 + 8.96T + 79T^{2} \) |
| 83 | \( 1 - 6.99T + 83T^{2} \) |
| 89 | \( 1 + 0.932T + 89T^{2} \) |
| 97 | \( 1 + 3.35T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.439847501700394283724626370535, −7.897335985906773551689715786625, −7.02897502083393427200363509827, −6.42491749936537339322016856389, −5.19067340222787098051356772817, −4.49348312716926288007413831436, −3.89233764457289730215758755295, −3.47875472106902916868512218885, −2.42518786887449764774251610784, −0.76364839070747289946839244006,
0.76364839070747289946839244006, 2.42518786887449764774251610784, 3.47875472106902916868512218885, 3.89233764457289730215758755295, 4.49348312716926288007413831436, 5.19067340222787098051356772817, 6.42491749936537339322016856389, 7.02897502083393427200363509827, 7.897335985906773551689715786625, 8.439847501700394283724626370535