| L(s) = 1 | + 5.72·2-s − 28.3·3-s + 0.821·4-s + 25·5-s − 162.·6-s + 40.2·7-s − 178.·8-s + 562.·9-s + 143.·10-s − 121·11-s − 23.3·12-s + 859.·13-s + 230.·14-s − 709.·15-s − 1.04e3·16-s − 172.·17-s + 3.22e3·18-s − 361·19-s + 20.5·20-s − 1.14e3·21-s − 693.·22-s + 3.60e3·23-s + 5.06e3·24-s + 625·25-s + 4.92e3·26-s − 9.06e3·27-s + 33.0·28-s + ⋯ |
| L(s) = 1 | + 1.01·2-s − 1.82·3-s + 0.0256·4-s + 0.447·5-s − 1.84·6-s + 0.310·7-s − 0.986·8-s + 2.31·9-s + 0.452·10-s − 0.301·11-s − 0.0467·12-s + 1.41·13-s + 0.314·14-s − 0.814·15-s − 1.02·16-s − 0.144·17-s + 2.34·18-s − 0.229·19-s + 0.0114·20-s − 0.565·21-s − 0.305·22-s + 1.42·23-s + 1.79·24-s + 0.200·25-s + 1.42·26-s − 2.39·27-s + 0.00797·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.783132600\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.783132600\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 25T \) |
| 11 | \( 1 + 121T \) |
| 19 | \( 1 + 361T \) |
| good | 2 | \( 1 - 5.72T + 32T^{2} \) |
| 3 | \( 1 + 28.3T + 243T^{2} \) |
| 7 | \( 1 - 40.2T + 1.68e4T^{2} \) |
| 13 | \( 1 - 859.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 172.T + 1.41e6T^{2} \) |
| 23 | \( 1 - 3.60e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 329.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.34e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.25e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 8.47e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 4.43e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 15.6T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.58e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.17e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.80e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.88e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.20e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.16e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.00e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.04e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.87e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 6.44e4T + 8.58e9T^{2} \) |
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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
−9.339381979793178948483976685945, −8.374483456371719088336514104027, −6.93055284227784960846634610887, −6.32452822322735056360543357095, −5.62535326265940798167728872025, −5.01471338303231266494935805271, −4.32875145166707236094503536211, −3.22360082792140473452856489657, −1.57692475546630677689059026239, −0.57380112048069305033590989428,
0.57380112048069305033590989428, 1.57692475546630677689059026239, 3.22360082792140473452856489657, 4.32875145166707236094503536211, 5.01471338303231266494935805271, 5.62535326265940798167728872025, 6.32452822322735056360543357095, 6.93055284227784960846634610887, 8.374483456371719088336514104027, 9.339381979793178948483976685945
