| L(s) = 1 | + 3-s + 3.16·5-s + 1.74·7-s + 9-s + 6.47·11-s − 1.41·13-s + 3.16·15-s + 2.47·17-s − 6.47·19-s + 1.74·21-s + 5.65·23-s + 5.00·25-s + 27-s + 5.99·29-s − 3.90·31-s + 6.47·33-s + 5.52·35-s − 10.5·37-s − 1.41·39-s − 2.47·41-s + 1.52·43-s + 3.16·45-s − 3.94·49-s + 2.47·51-s − 11.6·53-s + 20.4·55-s − 6.47·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.41·5-s + 0.660·7-s + 0.333·9-s + 1.95·11-s − 0.392·13-s + 0.816·15-s + 0.599·17-s − 1.48·19-s + 0.381·21-s + 1.17·23-s + 1.00·25-s + 0.192·27-s + 1.11·29-s − 0.702·31-s + 1.12·33-s + 0.934·35-s − 1.73·37-s − 0.226·39-s − 0.386·41-s + 0.232·43-s + 0.471·45-s − 0.563·49-s + 0.346·51-s − 1.59·53-s + 2.75·55-s − 0.857·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.705126219\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.705126219\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| good | 5 | \( 1 - 3.16T + 5T^{2} \) |
| 7 | \( 1 - 1.74T + 7T^{2} \) |
| 11 | \( 1 - 6.47T + 11T^{2} \) |
| 13 | \( 1 + 1.41T + 13T^{2} \) |
| 17 | \( 1 - 2.47T + 17T^{2} \) |
| 19 | \( 1 + 6.47T + 19T^{2} \) |
| 23 | \( 1 - 5.65T + 23T^{2} \) |
| 29 | \( 1 - 5.99T + 29T^{2} \) |
| 31 | \( 1 + 3.90T + 31T^{2} \) |
| 37 | \( 1 + 10.5T + 37T^{2} \) |
| 41 | \( 1 + 2.47T + 41T^{2} \) |
| 43 | \( 1 - 1.52T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 - 8.94T + 59T^{2} \) |
| 61 | \( 1 + 2.08T + 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 + 9.15T + 71T^{2} \) |
| 73 | \( 1 - 2.94T + 73T^{2} \) |
| 79 | \( 1 + 7.40T + 79T^{2} \) |
| 83 | \( 1 + 6.47T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
| 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
−8.792900171576327402199175756283, −8.220884279890624647310831515565, −6.89185289824628297123971375699, −6.67018568550055715035794990642, −5.66946054496783088609642736698, −4.84381195408923648268449314870, −3.99948753103343983426563002597, −2.96282212973609135692457227542, −1.86632529778747746872437235824, −1.36675926702826513573526540961,
1.36675926702826513573526540961, 1.86632529778747746872437235824, 2.96282212973609135692457227542, 3.99948753103343983426563002597, 4.84381195408923648268449314870, 5.66946054496783088609642736698, 6.67018568550055715035794990642, 6.89185289824628297123971375699, 8.220884279890624647310831515565, 8.792900171576327402199175756283
