| L(s) = 1 | + 77.9·2-s + 252.·3-s + 4.03e3·4-s + 9.43e3·5-s + 1.96e4·6-s − 4.57e4·7-s + 1.54e5·8-s − 1.13e5·9-s + 7.35e5·10-s − 1.38e5·11-s + 1.01e6·12-s + 1.42e6·13-s − 3.57e6·14-s + 2.38e6·15-s + 3.80e6·16-s − 1.41e6·17-s − 8.84e6·18-s − 8.80e6·19-s + 3.80e7·20-s − 1.15e7·21-s − 1.07e7·22-s + 4.59e7·23-s + 3.90e7·24-s + 4.02e7·25-s + 1.11e8·26-s − 7.33e7·27-s − 1.84e8·28-s + ⋯ |
| L(s) = 1 | + 1.72·2-s + 0.599·3-s + 1.96·4-s + 1.35·5-s + 1.03·6-s − 1.02·7-s + 1.66·8-s − 0.640·9-s + 2.32·10-s − 0.259·11-s + 1.18·12-s + 1.06·13-s − 1.77·14-s + 0.810·15-s + 0.907·16-s − 0.242·17-s − 1.10·18-s − 0.815·19-s + 2.65·20-s − 0.617·21-s − 0.446·22-s + 1.48·23-s + 1.00·24-s + 0.824·25-s + 1.83·26-s − 0.983·27-s − 2.02·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(5.889745770\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.889745770\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + 1.41e6T \) |
| good | 2 | \( 1 - 77.9T + 2.04e3T^{2} \) |
| 3 | \( 1 - 252.T + 1.77e5T^{2} \) |
| 5 | \( 1 - 9.43e3T + 4.88e7T^{2} \) |
| 7 | \( 1 + 4.57e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 1.38e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 1.42e6T + 1.79e12T^{2} \) |
| 19 | \( 1 + 8.80e6T + 1.16e14T^{2} \) |
| 23 | \( 1 - 4.59e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 1.77e8T + 1.22e16T^{2} \) |
| 31 | \( 1 - 1.08e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 7.31e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 9.38e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 2.51e8T + 9.29e17T^{2} \) |
| 47 | \( 1 - 1.08e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 3.62e9T + 9.26e18T^{2} \) |
| 59 | \( 1 - 8.63e9T + 3.01e19T^{2} \) |
| 61 | \( 1 + 1.36e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 1.02e10T + 1.22e20T^{2} \) |
| 71 | \( 1 - 1.96e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 4.93e9T + 3.13e20T^{2} \) |
| 79 | \( 1 + 3.26e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 4.10e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 7.31e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 1.24e10T + 7.15e21T^{2} \) |
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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
−15.69521835217434319123517776433, −14.45123665553572294337068136237, −13.44684183976661127003532530945, −12.90185550834934295714244276415, −10.93347745799948221606714629355, −9.132613897029473535871702524287, −6.52696090975166468302931009930, −5.53657926156264948823905728047, −3.46309977093492998297261707239, −2.26885636255484030206369633002,
2.26885636255484030206369633002, 3.46309977093492998297261707239, 5.53657926156264948823905728047, 6.52696090975166468302931009930, 9.132613897029473535871702524287, 10.93347745799948221606714629355, 12.90185550834934295714244276415, 13.44684183976661127003532530945, 14.45123665553572294337068136237, 15.69521835217434319123517776433
