| L(s) = 1 | + 1.49·3-s + 84.8·5-s − 2.30·7-s − 240.·9-s + 516.·11-s + 242.·13-s + 127.·15-s − 1.16e3·17-s − 1.20e3·19-s − 3.45·21-s − 1.97e3·23-s + 4.07e3·25-s − 725.·27-s + 1.75e3·29-s − 2.73e3·31-s + 774.·33-s − 195.·35-s − 1.21e4·37-s + 364.·39-s + 5.29e3·41-s − 3.25e3·43-s − 2.04e4·45-s − 5.83e3·47-s − 1.68e4·49-s − 1.75e3·51-s − 1.69e4·53-s + 4.38e4·55-s + ⋯ |
| L(s) = 1 | + 0.0962·3-s + 1.51·5-s − 0.0177·7-s − 0.990·9-s + 1.28·11-s + 0.398·13-s + 0.146·15-s − 0.979·17-s − 0.763·19-s − 0.00170·21-s − 0.777·23-s + 1.30·25-s − 0.191·27-s + 0.388·29-s − 0.511·31-s + 0.123·33-s − 0.0269·35-s − 1.45·37-s + 0.0383·39-s + 0.491·41-s − 0.268·43-s − 1.50·45-s − 0.385·47-s − 0.999·49-s − 0.0942·51-s − 0.830·53-s + 1.95·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 257 | \( 1 - 6.60e4T \) |
| good | 3 | \( 1 - 1.49T + 243T^{2} \) |
| 5 | \( 1 - 84.8T + 3.12e3T^{2} \) |
| 7 | \( 1 + 2.30T + 1.68e4T^{2} \) |
| 11 | \( 1 - 516.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 242.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.16e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.20e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.97e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.75e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.73e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.21e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 5.29e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 3.25e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 5.83e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.69e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 8.47e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.28e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.51e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.35e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.25e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.93e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.35e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.28e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.11e5T + 8.58e9T^{2} \) |
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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
−8.970101912340464046921711322627, −8.232504730743099102481488850432, −6.73361299687981423328775761684, −6.27718813779684994761313871457, −5.55570476958946583064766302877, −4.44668083891196574081169185478, −3.30193038528723412696406494242, −2.16817283654315655017901876827, −1.50290385569933269754292301330, 0,
1.50290385569933269754292301330, 2.16817283654315655017901876827, 3.30193038528723412696406494242, 4.44668083891196574081169185478, 5.55570476958946583064766302877, 6.27718813779684994761313871457, 6.73361299687981423328775761684, 8.232504730743099102481488850432, 8.970101912340464046921711322627
