| L(s) = 1 | + 8.27·2-s + 25.6·3-s + 36.4·4-s − 28.7·5-s + 212.·6-s + 37.0·8-s + 414.·9-s − 237.·10-s − 270.·11-s + 935.·12-s − 300.·13-s − 737.·15-s − 860.·16-s − 613.·17-s + 3.43e3·18-s + 1.70e3·19-s − 1.04e3·20-s − 2.23e3·22-s + 3.18e3·23-s + 949.·24-s − 2.29e3·25-s − 2.48e3·26-s + 4.40e3·27-s + 4.29e3·29-s − 6.10e3·30-s − 2.02e3·31-s − 8.30e3·32-s + ⋯ |
| L(s) = 1 | + 1.46·2-s + 1.64·3-s + 1.13·4-s − 0.514·5-s + 2.40·6-s + 0.204·8-s + 1.70·9-s − 0.752·10-s − 0.673·11-s + 1.87·12-s − 0.493·13-s − 0.846·15-s − 0.840·16-s − 0.514·17-s + 2.49·18-s + 1.08·19-s − 0.586·20-s − 0.984·22-s + 1.25·23-s + 0.336·24-s − 0.735·25-s − 0.721·26-s + 1.16·27-s + 0.949·29-s − 1.23·30-s − 0.379·31-s − 1.43·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.800578530\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.800578530\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 - 8.27T + 32T^{2} \) |
| 3 | \( 1 - 25.6T + 243T^{2} \) |
| 5 | \( 1 + 28.7T + 3.12e3T^{2} \) |
| 11 | \( 1 + 270.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 300.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 613.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.70e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.18e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.29e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.02e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.15e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.14e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.95e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.99e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.94e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.97e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.05e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.05e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.28e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.11e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.18e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.18e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.16e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 4.36e4T + 8.58e9T^{2} \) |
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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
−14.50650931889253454991758887742, −13.53836852114974685457043429737, −12.85004302132542152400669724060, −11.50487530191754428491634566447, −9.651982442170722786160835903192, −8.289595929222964735770111160818, −7.06386321099988479912435345948, −4.97034185271461722802199054572, −3.60370576816232366154835316369, −2.56382232392017197037896481000,
2.56382232392017197037896481000, 3.60370576816232366154835316369, 4.97034185271461722802199054572, 7.06386321099988479912435345948, 8.289595929222964735770111160818, 9.651982442170722786160835903192, 11.50487530191754428491634566447, 12.85004302132542152400669724060, 13.53836852114974685457043429737, 14.50650931889253454991758887742
