L(s) = 1 | − 10.6·2-s − 22.7·3-s + 81.4·4-s − 25·5-s + 242.·6-s − 527.·8-s + 273.·9-s + 266.·10-s + 521.·11-s − 1.85e3·12-s + 530.·13-s + 567.·15-s + 3.00e3·16-s + 1.99e3·17-s − 2.90e3·18-s + 2.49e3·19-s − 2.03e3·20-s − 5.55e3·22-s − 3.55e3·23-s + 1.19e4·24-s + 625·25-s − 5.65e3·26-s − 683.·27-s − 4.28e3·29-s − 6.05e3·30-s + 5.03e3·31-s − 1.51e4·32-s + ⋯ |
L(s) = 1 | − 1.88·2-s − 1.45·3-s + 2.54·4-s − 0.447·5-s + 2.74·6-s − 2.91·8-s + 1.12·9-s + 0.842·10-s + 1.29·11-s − 3.71·12-s + 0.870·13-s + 0.651·15-s + 2.93·16-s + 1.67·17-s − 2.11·18-s + 1.58·19-s − 1.13·20-s − 2.44·22-s − 1.40·23-s + 4.24·24-s + 0.200·25-s − 1.63·26-s − 0.180·27-s − 0.946·29-s − 1.22·30-s + 0.941·31-s − 2.62·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.5199985229\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5199985229\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + 25T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 10.6T + 32T^{2} \) |
| 3 | \( 1 + 22.7T + 243T^{2} \) |
| 11 | \( 1 - 521.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 530.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.99e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.49e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.55e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.28e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.03e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 2.62e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.82e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 5.08e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 4.98e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.90e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 8.93e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.35e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.37e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.69e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 9.79e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.85e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.55e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.12e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.56e5T + 8.58e9T^{2} \) |
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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
−11.20014895088847964437218967129, −10.16893879237515443993369326405, −9.523141077569923203464357446599, −8.295397614363897938816072889267, −7.38324057593702539078079870166, −6.41435932055632083811359868655, −5.60810969459520444745431334988, −3.56340642120597535605975324378, −1.38634450490750344577128333820, −0.69396773514021966860137750197,
0.69396773514021966860137750197, 1.38634450490750344577128333820, 3.56340642120597535605975324378, 5.60810969459520444745431334988, 6.41435932055632083811359868655, 7.38324057593702539078079870166, 8.295397614363897938816072889267, 9.523141077569923203464357446599, 10.16893879237515443993369326405, 11.20014895088847964437218967129