| L(s) = 1 | + 29.5·3-s + 41.2·5-s − 77.0·7-s + 632.·9-s + 121·11-s − 637.·13-s + 1.22e3·15-s + 1.42e3·17-s + 1.71e3·19-s − 2.28e3·21-s + 4.19e3·23-s − 1.42e3·25-s + 1.15e4·27-s + 878.·29-s − 6.76e3·31-s + 3.58e3·33-s − 3.17e3·35-s − 4.00e3·37-s − 1.88e4·39-s − 1.36e4·41-s − 1.62e3·43-s + 2.60e4·45-s + 1.71e4·47-s − 1.08e4·49-s + 4.21e4·51-s + 1.51e4·53-s + 4.98e3·55-s + ⋯ |
| L(s) = 1 | + 1.89·3-s + 0.737·5-s − 0.594·7-s + 2.60·9-s + 0.301·11-s − 1.04·13-s + 1.40·15-s + 1.19·17-s + 1.09·19-s − 1.12·21-s + 1.65·23-s − 0.455·25-s + 3.04·27-s + 0.193·29-s − 1.26·31-s + 0.572·33-s − 0.438·35-s − 0.481·37-s − 1.98·39-s − 1.27·41-s − 0.134·43-s + 1.91·45-s + 1.13·47-s − 0.646·49-s + 2.26·51-s + 0.740·53-s + 0.222·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.380518885\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.380518885\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - 121T \) |
| good | 3 | \( 1 - 29.5T + 243T^{2} \) |
| 5 | \( 1 - 41.2T + 3.12e3T^{2} \) |
| 7 | \( 1 + 77.0T + 1.68e4T^{2} \) |
| 13 | \( 1 + 637.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.42e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.71e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.19e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 878.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.76e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.00e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.36e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.62e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.71e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.51e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.22e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.52e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.84e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.56e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.96e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.07e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.29e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.07e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.44e4T + 8.58e9T^{2} \) |
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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
−12.16439923588664503455133900016, −10.27104608776563147031566061527, −9.558287216805528934135622477954, −9.018833966861268827972839676658, −7.70152003114787942435264094707, −6.95871997629458097131436093722, −5.21322987011102319725784400200, −3.56514418249012423677702258015, −2.74042401048019756521052565843, −1.45557761468651860679098532501,
1.45557761468651860679098532501, 2.74042401048019756521052565843, 3.56514418249012423677702258015, 5.21322987011102319725784400200, 6.95871997629458097131436093722, 7.70152003114787942435264094707, 9.018833966861268827972839676658, 9.558287216805528934135622477954, 10.27104608776563147031566061527, 12.16439923588664503455133900016
