| L(s) = 1 | − 2.30·3-s − 5-s − 7-s + 2.30·9-s + 1.30·11-s + 3.60·13-s + 2.30·15-s + 5.30·17-s + 19-s + 2.30·21-s − 7.60·23-s − 4·25-s + 1.60·27-s − 0.697·29-s − 4.69·31-s − 3·33-s + 35-s − 3.60·37-s − 8.30·39-s − 3.30·41-s − 7.21·43-s − 2.30·45-s − 0.394·47-s + 49-s − 12.2·51-s + 6.90·53-s − 1.30·55-s + ⋯ |
| L(s) = 1 | − 1.32·3-s − 0.447·5-s − 0.377·7-s + 0.767·9-s + 0.392·11-s + 1.00·13-s + 0.594·15-s + 1.28·17-s + 0.229·19-s + 0.502·21-s − 1.58·23-s − 0.800·25-s + 0.308·27-s − 0.129·29-s − 0.843·31-s − 0.522·33-s + 0.169·35-s − 0.592·37-s − 1.32·39-s − 0.515·41-s − 1.09·43-s − 0.343·45-s − 0.0575·47-s + 0.142·49-s − 1.70·51-s + 0.948·53-s − 0.175·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1064 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1064 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 2.30T + 3T^{2} \) |
| 5 | \( 1 + T + 5T^{2} \) |
| 11 | \( 1 - 1.30T + 11T^{2} \) |
| 13 | \( 1 - 3.60T + 13T^{2} \) |
| 17 | \( 1 - 5.30T + 17T^{2} \) |
| 23 | \( 1 + 7.60T + 23T^{2} \) |
| 29 | \( 1 + 0.697T + 29T^{2} \) |
| 31 | \( 1 + 4.69T + 31T^{2} \) |
| 37 | \( 1 + 3.60T + 37T^{2} \) |
| 41 | \( 1 + 3.30T + 41T^{2} \) |
| 43 | \( 1 + 7.21T + 43T^{2} \) |
| 47 | \( 1 + 0.394T + 47T^{2} \) |
| 53 | \( 1 - 6.90T + 53T^{2} \) |
| 59 | \( 1 + 1.60T + 59T^{2} \) |
| 61 | \( 1 + 6.21T + 61T^{2} \) |
| 67 | \( 1 + 0.0916T + 67T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 73 | \( 1 - 4.90T + 73T^{2} \) |
| 79 | \( 1 + 14T + 79T^{2} \) |
| 83 | \( 1 - 7.51T + 83T^{2} \) |
| 89 | \( 1 + 7.81T + 89T^{2} \) |
| 97 | \( 1 + 4.81T + 97T^{2} \) |
| 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
−9.745117359565659357242330985903, −8.603365665371041337773328325363, −7.73644267021588761653508413257, −6.77689653930607795693004811791, −5.93038822236590542551801023854, −5.45696108401680629619920110978, −4.16623060970997008570478854673, −3.38522443720405232099882162259, −1.47341862756899216645521327926, 0,
1.47341862756899216645521327926, 3.38522443720405232099882162259, 4.16623060970997008570478854673, 5.45696108401680629619920110978, 5.93038822236590542551801023854, 6.77689653930607795693004811791, 7.73644267021588761653508413257, 8.603365665371041337773328325363, 9.745117359565659357242330985903
