| L(s) = 1 | − 0.554i·3-s − 4.57·5-s + 2.64i·7-s + 8.69·9-s + 15.7i·11-s + 8.57·13-s + 2.53i·15-s + 28.3·17-s + 6.33i·19-s + 1.46·21-s + 31.0i·23-s − 4.05·25-s − 9.81i·27-s − 0.846·29-s − 21.6i·31-s + ⋯ |
| L(s) = 1 | − 0.184i·3-s − 0.915·5-s + 0.377i·7-s + 0.965·9-s + 1.43i·11-s + 0.659·13-s + 0.169i·15-s + 1.66·17-s + 0.333i·19-s + 0.0698·21-s + 1.35i·23-s − 0.162·25-s − 0.363i·27-s − 0.0291·29-s − 0.697i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.28660 + 0.532928i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.28660 + 0.532928i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 2.64iT \) |
| good | 3 | \( 1 + 0.554iT - 9T^{2} \) |
| 5 | \( 1 + 4.57T + 25T^{2} \) |
| 11 | \( 1 - 15.7iT - 121T^{2} \) |
| 13 | \( 1 - 8.57T + 169T^{2} \) |
| 17 | \( 1 - 28.3T + 289T^{2} \) |
| 19 | \( 1 - 6.33iT - 361T^{2} \) |
| 23 | \( 1 - 31.0iT - 529T^{2} \) |
| 29 | \( 1 + 0.846T + 841T^{2} \) |
| 31 | \( 1 + 21.6iT - 961T^{2} \) |
| 37 | \( 1 + 33.6T + 1.36e3T^{2} \) |
| 41 | \( 1 - 66.9T + 1.68e3T^{2} \) |
| 43 | \( 1 - 44.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 38.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 14.8T + 2.80e3T^{2} \) |
| 59 | \( 1 - 5.80iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 52.6T + 3.72e3T^{2} \) |
| 67 | \( 1 + 117. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 81.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 47.8T + 5.32e3T^{2} \) |
| 79 | \( 1 - 57.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 102. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 89.2T + 7.92e3T^{2} \) |
| 97 | \( 1 + 3.44T + 9.40e3T^{2} \) |
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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
−12.27280707137665353767534448827, −11.36521551227446917082030615839, −10.09968730781178910315350606110, −9.363268678604661081338668404457, −7.73334466946236490577582336164, −7.52048340797495048801694595620, −5.99845179738711105366692464226, −4.59489274929040809600377695679, −3.51682571060593413834509512078, −1.55485861189045533252588261315,
0.885762572433443073667195869933, 3.32376393142946992140980011745, 4.18440325178984026885294169897, 5.64000370532332327092665889492, 6.95674587664764970930380862472, 7.952885458650635550347452249367, 8.803070589439741883471218967175, 10.18799622410495823920758640028, 10.87735111335992321386909772328, 11.88300074730888753813002409486
