| L(s) = 1 | − 1.04·2-s − 0.911·4-s − 1.64·5-s + 5.01·7-s + 3.03·8-s + 1.71·10-s − 11-s − 3.01·13-s − 5.22·14-s − 1.34·16-s + 3.18·17-s − 6.03·19-s + 1.50·20-s + 1.04·22-s + 4.89·23-s − 2.28·25-s + 3.14·26-s − 4.56·28-s + 0.904·29-s + 5.23·31-s − 4.67·32-s − 3.32·34-s − 8.26·35-s − 5.38·37-s + 6.29·38-s − 5.00·40-s + 11.7·41-s + ⋯ |
| L(s) = 1 | − 0.737·2-s − 0.455·4-s − 0.737·5-s + 1.89·7-s + 1.07·8-s + 0.543·10-s − 0.301·11-s − 0.837·13-s − 1.39·14-s − 0.336·16-s + 0.773·17-s − 1.38·19-s + 0.336·20-s + 0.222·22-s + 1.02·23-s − 0.456·25-s + 0.617·26-s − 0.863·28-s + 0.167·29-s + 0.940·31-s − 0.825·32-s − 0.570·34-s − 1.39·35-s − 0.884·37-s + 1.02·38-s − 0.791·40-s + 1.82·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.068084594\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.068084594\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 + 1.04T + 2T^{2} \) |
| 5 | \( 1 + 1.64T + 5T^{2} \) |
| 7 | \( 1 - 5.01T + 7T^{2} \) |
| 13 | \( 1 + 3.01T + 13T^{2} \) |
| 17 | \( 1 - 3.18T + 17T^{2} \) |
| 19 | \( 1 + 6.03T + 19T^{2} \) |
| 23 | \( 1 - 4.89T + 23T^{2} \) |
| 29 | \( 1 - 0.904T + 29T^{2} \) |
| 31 | \( 1 - 5.23T + 31T^{2} \) |
| 37 | \( 1 + 5.38T + 37T^{2} \) |
| 41 | \( 1 - 11.7T + 41T^{2} \) |
| 43 | \( 1 + 11.2T + 43T^{2} \) |
| 47 | \( 1 - 7.10T + 47T^{2} \) |
| 53 | \( 1 - 13.0T + 53T^{2} \) |
| 59 | \( 1 + 0.468T + 59T^{2} \) |
| 67 | \( 1 + 3.15T + 67T^{2} \) |
| 71 | \( 1 + 9.71T + 71T^{2} \) |
| 73 | \( 1 + 3.32T + 73T^{2} \) |
| 79 | \( 1 + 5.59T + 79T^{2} \) |
| 83 | \( 1 - 0.444T + 83T^{2} \) |
| 89 | \( 1 + 3.27T + 89T^{2} \) |
| 97 | \( 1 - 7.38T + 97T^{2} \) |
| 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
−8.306081065784029726865887655230, −7.44492381425276954727070019691, −7.24249977540781349823069129134, −5.79090533357716933993190317404, −4.96782961663243749998643749791, −4.55024681554024636517697466565, −3.89386397148024531004711402346, −2.53878293783162887365977672039, −1.61502017957796694452010998310, −0.64274654712239766876475651204,
0.64274654712239766876475651204, 1.61502017957796694452010998310, 2.53878293783162887365977672039, 3.89386397148024531004711402346, 4.55024681554024636517697466565, 4.96782961663243749998643749791, 5.79090533357716933993190317404, 7.24249977540781349823069129134, 7.44492381425276954727070019691, 8.306081065784029726865887655230
