| L(s) = 1 | − 0.984·2-s + 1.73·3-s − 3.03·4-s − 2.95i·5-s − 1.70·6-s + 2.64i·7-s + 6.91·8-s + 2.99·9-s + 2.91i·10-s + 1.64i·11-s − 5.25·12-s − 20.6·13-s − 2.60i·14-s − 5.12i·15-s + 5.31·16-s + 1.80i·17-s + ⋯ |
| L(s) = 1 | − 0.492·2-s + 0.577·3-s − 0.757·4-s − 0.591i·5-s − 0.284·6-s + 0.377i·7-s + 0.864·8-s + 0.333·9-s + 0.291i·10-s + 0.149i·11-s − 0.437·12-s − 1.58·13-s − 0.185i·14-s − 0.341i·15-s + 0.332·16-s + 0.106i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 - 0.200i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.979 - 0.200i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.006802771624\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.006802771624\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 1.73T \) |
| 7 | \( 1 - 2.64iT \) |
| 23 | \( 1 + (4.61 - 22.5i)T \) |
| good | 2 | \( 1 + 0.984T + 4T^{2} \) |
| 5 | \( 1 + 2.95iT - 25T^{2} \) |
| 11 | \( 1 - 1.64iT - 121T^{2} \) |
| 13 | \( 1 + 20.6T + 169T^{2} \) |
| 17 | \( 1 - 1.80iT - 289T^{2} \) |
| 19 | \( 1 + 30.5iT - 361T^{2} \) |
| 29 | \( 1 + 23.0T + 841T^{2} \) |
| 31 | \( 1 + 46.3T + 961T^{2} \) |
| 37 | \( 1 - 61.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 46.0T + 1.68e3T^{2} \) |
| 43 | \( 1 - 47.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 77.1T + 2.20e3T^{2} \) |
| 53 | \( 1 + 75.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 59.3T + 3.48e3T^{2} \) |
| 61 | \( 1 + 13.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 24.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 24.4T + 5.04e3T^{2} \) |
| 73 | \( 1 + 5.03T + 5.32e3T^{2} \) |
| 79 | \( 1 + 120. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 84.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 62.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 49.7iT - 9.40e3T^{2} \) |
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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
−11.08429527950794124134206916632, −9.755334812918075526876101537847, −9.473858624215313780108587392304, −8.632189965625352981490990496368, −7.80547473430492519376896119792, −6.92216162665573953992695156110, −5.13555752060625131960487022283, −4.71902673743281691485566320792, −3.21572502277770580469742616645, −1.72526143392981970156427337009,
0.00300831279112797365516213997, 1.91448250031726429109686698545, 3.38350823577132294086697434302, 4.39305022096838214647645009555, 5.55265568225939261945130921530, 7.09531371638641124322038303506, 7.65003780968628083199729702233, 8.611000945728733094382481918367, 9.505649019472820800870335974827, 10.19416725999346437328548031567
