| L(s) = 1 | + 1.61·3-s − 0.618·5-s + 7-s − 0.381·9-s + 0.763·11-s + 1.23·13-s − 1.00·15-s − 8.47·19-s + 1.61·21-s + 7.70·23-s − 4.61·25-s − 5.47·27-s + 5.70·29-s − 6.32·31-s + 1.23·33-s − 0.618·35-s − 0.472·37-s + 2.00·39-s + 0.0901·41-s − 12.0·43-s + 0.236·45-s − 8.47·47-s + 49-s + 10.7·53-s − 0.472·55-s − 13.7·57-s + 7.32·61-s + ⋯ |
| L(s) = 1 | + 0.934·3-s − 0.276·5-s + 0.377·7-s − 0.127·9-s + 0.230·11-s + 0.342·13-s − 0.258·15-s − 1.94·19-s + 0.353·21-s + 1.60·23-s − 0.923·25-s − 1.05·27-s + 1.05·29-s − 1.13·31-s + 0.215·33-s − 0.104·35-s − 0.0776·37-s + 0.320·39-s + 0.0140·41-s − 1.84·43-s + 0.0351·45-s − 1.23·47-s + 0.142·49-s + 1.48·53-s − 0.0636·55-s − 1.81·57-s + 0.938·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8092 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8092 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - 1.61T + 3T^{2} \) |
| 5 | \( 1 + 0.618T + 5T^{2} \) |
| 11 | \( 1 - 0.763T + 11T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 19 | \( 1 + 8.47T + 19T^{2} \) |
| 23 | \( 1 - 7.70T + 23T^{2} \) |
| 29 | \( 1 - 5.70T + 29T^{2} \) |
| 31 | \( 1 + 6.32T + 31T^{2} \) |
| 37 | \( 1 + 0.472T + 37T^{2} \) |
| 41 | \( 1 - 0.0901T + 41T^{2} \) |
| 43 | \( 1 + 12.0T + 43T^{2} \) |
| 47 | \( 1 + 8.47T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 7.32T + 61T^{2} \) |
| 67 | \( 1 + 13.0T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 + 7.14T + 73T^{2} \) |
| 79 | \( 1 + 2.94T + 79T^{2} \) |
| 83 | \( 1 - 15.4T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 15.0T + 97T^{2} \) |
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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
−7.60217189889495518950797540861, −6.87224664566726037575593040970, −6.21010520051678210091185781284, −5.32200943795411119119447140587, −4.51853825561178375159393747385, −3.80365177339396365432613331841, −3.11695610449708059222826533597, −2.27641201960097222341600664793, −1.48498691625746550509108414318, 0,
1.48498691625746550509108414318, 2.27641201960097222341600664793, 3.11695610449708059222826533597, 3.80365177339396365432613331841, 4.51853825561178375159393747385, 5.32200943795411119119447140587, 6.21010520051678210091185781284, 6.87224664566726037575593040970, 7.60217189889495518950797540861
