| L(s) = 1 | + 5.35·2-s + 20.7·4-s − 4.43·7-s + 68.1·8-s + 3.43·11-s + 78.7·13-s − 23.7·14-s + 199.·16-s − 53.1·17-s + 20.4·19-s + 18.4·22-s − 118.·23-s + 421.·26-s − 91.8·28-s − 168.·29-s − 61.0·31-s + 523.·32-s − 284.·34-s − 246.·37-s + 109.·38-s − 422.·41-s + 362.·43-s + 71.1·44-s − 633.·46-s + 170.·47-s − 323.·49-s + 1.63e3·52-s + ⋯ |
| L(s) = 1 | + 1.89·2-s + 2.58·4-s − 0.239·7-s + 3.01·8-s + 0.0941·11-s + 1.67·13-s − 0.453·14-s + 3.11·16-s − 0.758·17-s + 0.246·19-s + 0.178·22-s − 1.07·23-s + 3.18·26-s − 0.620·28-s − 1.07·29-s − 0.353·31-s + 2.89·32-s − 1.43·34-s − 1.09·37-s + 0.467·38-s − 1.60·41-s + 1.28·43-s + 0.243·44-s − 2.03·46-s + 0.529·47-s − 0.942·49-s + 4.35·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.618296482\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.618296482\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 5.35T + 8T^{2} \) |
| 7 | \( 1 + 4.43T + 343T^{2} \) |
| 11 | \( 1 - 3.43T + 1.33e3T^{2} \) |
| 13 | \( 1 - 78.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 53.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 20.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 118.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 168.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 61.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 246.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 422.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 362.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 170.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 546.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 216.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 130.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 614.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 324.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 88.8T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.13e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 758.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 195.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 521T + 9.12e5T^{2} \) |
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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
−11.95851388148898864194797993291, −11.23468258661010407582143322923, −10.36104681288976294575126699450, −8.698487455767657844817834672950, −7.28844657601254109514013843367, −6.28694315425350630600039144592, −5.53725700464736748151346605983, −4.17703267214602182631512056044, −3.39149222797908716005704411854, −1.85549412255055055166707447739,
1.85549412255055055166707447739, 3.39149222797908716005704411854, 4.17703267214602182631512056044, 5.53725700464736748151346605983, 6.28694315425350630600039144592, 7.28844657601254109514013843367, 8.698487455767657844817834672950, 10.36104681288976294575126699450, 11.23468258661010407582143322923, 11.95851388148898864194797993291
