| L(s) = 1 | + 0.414·2-s + 1.41·3-s − 1.82·4-s + 5-s + 0.585·6-s − 0.828·7-s − 1.58·8-s − 0.999·9-s + 0.414·10-s + 0.585·11-s − 2.58·12-s − 13-s − 0.343·14-s + 1.41·15-s + 3·16-s − 4.82·17-s − 0.414·18-s + 3.41·19-s − 1.82·20-s − 1.17·21-s + 0.242·22-s − 1.41·23-s − 2.24·24-s + 25-s − 0.414·26-s − 5.65·27-s + 1.51·28-s + ⋯ |
| L(s) = 1 | + 0.292·2-s + 0.816·3-s − 0.914·4-s + 0.447·5-s + 0.239·6-s − 0.313·7-s − 0.560·8-s − 0.333·9-s + 0.130·10-s + 0.176·11-s − 0.746·12-s − 0.277·13-s − 0.0917·14-s + 0.365·15-s + 0.750·16-s − 1.17·17-s − 0.0976·18-s + 0.783·19-s − 0.408·20-s − 0.255·21-s + 0.0517·22-s − 0.294·23-s − 0.457·24-s + 0.200·25-s − 0.0812·26-s − 1.08·27-s + 0.286·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.063070829\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.063070829\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 0.414T + 2T^{2} \) |
| 3 | \( 1 - 1.41T + 3T^{2} \) |
| 7 | \( 1 + 0.828T + 7T^{2} \) |
| 11 | \( 1 - 0.585T + 11T^{2} \) |
| 17 | \( 1 + 4.82T + 17T^{2} \) |
| 19 | \( 1 - 3.41T + 19T^{2} \) |
| 23 | \( 1 + 1.41T + 23T^{2} \) |
| 29 | \( 1 - 5.65T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 - 8.48T + 37T^{2} \) |
| 41 | \( 1 + 8.82T + 41T^{2} \) |
| 43 | \( 1 - 3.07T + 43T^{2} \) |
| 47 | \( 1 - 0.828T + 47T^{2} \) |
| 53 | \( 1 + 14.4T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 + 2T + 67T^{2} \) |
| 71 | \( 1 + 7.89T + 71T^{2} \) |
| 73 | \( 1 + 8.48T + 73T^{2} \) |
| 79 | \( 1 - 8.48T + 79T^{2} \) |
| 83 | \( 1 + 8.82T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 3.65T + 97T^{2} \) |
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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
−14.56910906359927134647186589325, −13.81112759251206052438010784372, −13.12594620823082647977800969405, −11.76834210574544440479386962876, −9.990855243749131559826814337555, −9.112435927615463317006778046129, −8.129012086973190396606820751602, −6.24284167971529424103249376325, −4.64045613514715764195979885236, −2.96910192867035627498111348957,
2.96910192867035627498111348957, 4.64045613514715764195979885236, 6.24284167971529424103249376325, 8.129012086973190396606820751602, 9.112435927615463317006778046129, 9.990855243749131559826814337555, 11.76834210574544440479386962876, 13.12594620823082647977800969405, 13.81112759251206052438010784372, 14.56910906359927134647186589325
