| L(s) = 1 | + 55.7·2-s + 800.·3-s + 1.05e3·4-s − 7.06e3·5-s + 4.46e4·6-s − 1.68e4·7-s − 5.52e4·8-s + 4.64e5·9-s − 3.93e5·10-s + 1.29e5·11-s + 8.46e5·12-s − 1.02e6·13-s − 9.36e5·14-s − 5.65e6·15-s − 5.24e6·16-s + 1.33e6·17-s + 2.58e7·18-s + 7.49e6·19-s − 7.47e6·20-s − 1.34e7·21-s + 7.23e6·22-s + 1.46e7·23-s − 4.42e7·24-s + 1.10e6·25-s − 5.72e7·26-s + 2.29e8·27-s − 1.77e7·28-s + ⋯ |
| L(s) = 1 | + 1.23·2-s + 1.90·3-s + 0.516·4-s − 1.01·5-s + 2.34·6-s − 0.377·7-s − 0.595·8-s + 2.62·9-s − 1.24·10-s + 0.243·11-s + 0.982·12-s − 0.767·13-s − 0.465·14-s − 1.92·15-s − 1.24·16-s + 0.227·17-s + 3.22·18-s + 0.694·19-s − 0.522·20-s − 0.719·21-s + 0.299·22-s + 0.475·23-s − 1.13·24-s + 0.0225·25-s − 0.944·26-s + 3.08·27-s − 0.195·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(3.741534807\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.741534807\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + 1.68e4T \) |
| good | 2 | \( 1 - 55.7T + 2.04e3T^{2} \) |
| 3 | \( 1 - 800.T + 1.77e5T^{2} \) |
| 5 | \( 1 + 7.06e3T + 4.88e7T^{2} \) |
| 11 | \( 1 - 1.29e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 1.02e6T + 1.79e12T^{2} \) |
| 17 | \( 1 - 1.33e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 7.49e6T + 1.16e14T^{2} \) |
| 23 | \( 1 - 1.46e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 1.55e8T + 1.22e16T^{2} \) |
| 31 | \( 1 - 2.01e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 2.58e7T + 1.77e17T^{2} \) |
| 41 | \( 1 - 5.33e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 6.21e8T + 9.29e17T^{2} \) |
| 47 | \( 1 - 2.05e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 1.57e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + 1.12e8T + 3.01e19T^{2} \) |
| 61 | \( 1 + 9.95e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 2.01e10T + 1.22e20T^{2} \) |
| 71 | \( 1 - 1.10e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 5.04e9T + 3.13e20T^{2} \) |
| 79 | \( 1 + 1.76e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 3.75e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 6.39e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 8.87e9T + 7.15e21T^{2} \) |
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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
−19.93186314724499453368083707417, −18.90864860502843531213434550667, −15.65786370082188780068604824516, −14.77739591748498810225391912774, −13.63662388402639273946717675614, −12.31471395766097859287726181258, −9.272375092596959973052351557364, −7.52340847051685921770351915983, −4.16125578005830046604543854061, −2.95263408191541634718456541899,
2.95263408191541634718456541899, 4.16125578005830046604543854061, 7.52340847051685921770351915983, 9.272375092596959973052351557364, 12.31471395766097859287726181258, 13.63662388402639273946717675614, 14.77739591748498810225391912774, 15.65786370082188780068604824516, 18.90864860502843531213434550667, 19.93186314724499453368083707417
