L(s) = 1 | − 1.30·2-s + 2.30·3-s − 0.302·4-s + 1.30·5-s − 3·6-s − 7-s + 3·8-s + 2.30·9-s − 1.69·10-s + 1.69·11-s − 0.697·12-s − 2.30·13-s + 1.30·14-s + 3·15-s − 3.30·16-s − 5.60·17-s − 3.00·18-s − 7·19-s − 0.394·20-s − 2.30·21-s − 2.21·22-s + 4.69·23-s + 6.90·24-s − 3.30·25-s + 3·26-s − 1.60·27-s + 0.302·28-s + ⋯ |
L(s) = 1 | − 0.921·2-s + 1.32·3-s − 0.151·4-s + 0.582·5-s − 1.22·6-s − 0.377·7-s + 1.06·8-s + 0.767·9-s − 0.536·10-s + 0.511·11-s − 0.201·12-s − 0.638·13-s + 0.348·14-s + 0.774·15-s − 0.825·16-s − 1.35·17-s − 0.707·18-s − 1.60·19-s − 0.0882·20-s − 0.502·21-s − 0.471·22-s + 0.979·23-s + 1.41·24-s − 0.660·25-s + 0.588·26-s − 0.308·27-s + 0.0572·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8224050385\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8224050385\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 73 | \( 1 - T \) |
good | 2 | \( 1 + 1.30T + 2T^{2} \) |
| 3 | \( 1 - 2.30T + 3T^{2} \) |
| 5 | \( 1 - 1.30T + 5T^{2} \) |
| 7 | \( 1 + T + 7T^{2} \) |
| 11 | \( 1 - 1.69T + 11T^{2} \) |
| 13 | \( 1 + 2.30T + 13T^{2} \) |
| 17 | \( 1 + 5.60T + 17T^{2} \) |
| 19 | \( 1 + 7T + 19T^{2} \) |
| 23 | \( 1 - 4.69T + 23T^{2} \) |
| 29 | \( 1 - 8.21T + 29T^{2} \) |
| 31 | \( 1 + 0.605T + 31T^{2} \) |
| 37 | \( 1 - 7.60T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 - 9T + 47T^{2} \) |
| 53 | \( 1 + 8.21T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 2.69T + 61T^{2} \) |
| 67 | \( 1 - 12.8T + 67T^{2} \) |
| 71 | \( 1 - 6.90T + 71T^{2} \) |
| 79 | \( 1 + 4.90T + 79T^{2} \) |
| 83 | \( 1 - 12.5T + 83T^{2} \) |
| 89 | \( 1 + 4.81T + 89T^{2} \) |
| 97 | \( 1 - 2.90T + 97T^{2} \) |
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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
−14.50350199733923038756179334344, −13.64832028481806944316760166387, −12.80964818245449182699867595010, −10.83068722341473055107727760735, −9.621328782484073169562313905267, −9.016243251200494593807958055995, −8.111173661069387653941008641711, −6.67511392041592030737607723858, −4.34665090980010480433814472594, −2.32362777020342723225021402452,
2.32362777020342723225021402452, 4.34665090980010480433814472594, 6.67511392041592030737607723858, 8.111173661069387653941008641711, 9.016243251200494593807958055995, 9.621328782484073169562313905267, 10.83068722341473055107727760735, 12.80964818245449182699867595010, 13.64832028481806944316760166387, 14.50350199733923038756179334344