L(s) = 1 | − 2.35i·3-s + 2.23·5-s + 5.25i·7-s + 3.47·9-s + 19.9i·11-s + 8.47·13-s − 5.25i·15-s + 11.8·17-s − 15.2i·19-s + 12.3·21-s + 0.555i·23-s + 5.00·25-s − 29.3i·27-s + 10.9·29-s + 8.29i·31-s + ⋯ |
L(s) = 1 | − 0.783i·3-s + 0.447·5-s + 0.751i·7-s + 0.385·9-s + 1.81i·11-s + 0.651·13-s − 0.350i·15-s + 0.699·17-s − 0.800i·19-s + 0.588·21-s + 0.0241i·23-s + 0.200·25-s − 1.08i·27-s + 0.377·29-s + 0.267i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.91098\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.91098\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - 2.23T \) |
good | 3 | \( 1 + 2.35iT - 9T^{2} \) |
| 7 | \( 1 - 5.25iT - 49T^{2} \) |
| 11 | \( 1 - 19.9iT - 121T^{2} \) |
| 13 | \( 1 - 8.47T + 169T^{2} \) |
| 17 | \( 1 - 11.8T + 289T^{2} \) |
| 19 | \( 1 + 15.2iT - 361T^{2} \) |
| 23 | \( 1 - 0.555iT - 529T^{2} \) |
| 29 | \( 1 - 10.9T + 841T^{2} \) |
| 31 | \( 1 - 8.29iT - 961T^{2} \) |
| 37 | \( 1 - 18.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + 14.5T + 1.68e3T^{2} \) |
| 43 | \( 1 - 22.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 53.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 66.3T + 2.80e3T^{2} \) |
| 59 | \( 1 - 17.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 90.1T + 3.72e3T^{2} \) |
| 67 | \( 1 - 50.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 80.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 5.55T + 5.32e3T^{2} \) |
| 79 | \( 1 - 13.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 76.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 111.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 92.8T + 9.40e3T^{2} \) |
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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
−11.70977965263851148549033097582, −10.32505548405223934907145265300, −9.602435901919320346446320655497, −8.563351119039810314976088633158, −7.39337877714976062220643810942, −6.71069908052304013691699814495, −5.55866043077923825439788431593, −4.37934627267238151441255396734, −2.54356508116371214221037638935, −1.45820388001025216795132040605,
1.10978418672310627673152156832, 3.25949058607515899923515365180, 4.10537406850871062200977991313, 5.47213909694509482470300960254, 6.34347873748410983685767497535, 7.70873577238893634256152387371, 8.697878226765609503617838884226, 9.687075207655485459382825705813, 10.55219770451529673364904304194, 11.03516672541767280429355018284