L(s) = 1 | + (−0.0586 + 0.0586i)5-s − 4.61·7-s + (5.36 + 5.36i)11-s + (−11.0 − 11.0i)13-s + 12.8·17-s + (2.63 − 2.63i)19-s + 16.3·23-s + 24.9i·25-s + (−26.0 − 26.0i)29-s + 20.2i·31-s + (0.270 − 0.270i)35-s + (−41.2 + 41.2i)37-s + 3.29i·41-s + (−0.786 − 0.786i)43-s + 79.7i·47-s + ⋯ |
L(s) = 1 | + (−0.0117 + 0.0117i)5-s − 0.659·7-s + (0.487 + 0.487i)11-s + (−0.850 − 0.850i)13-s + 0.757·17-s + (0.138 − 0.138i)19-s + 0.712·23-s + 0.999i·25-s + (−0.898 − 0.898i)29-s + 0.652i·31-s + (0.00773 − 0.00773i)35-s + (−1.11 + 1.11i)37-s + 0.0804i·41-s + (−0.0183 − 0.0183i)43-s + 1.69i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.548 - 0.835i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.548 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8014144907\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8014144907\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.0586 - 0.0586i)T - 25iT^{2} \) |
| 7 | \( 1 + 4.61T + 49T^{2} \) |
| 11 | \( 1 + (-5.36 - 5.36i)T + 121iT^{2} \) |
| 13 | \( 1 + (11.0 + 11.0i)T + 169iT^{2} \) |
| 17 | \( 1 - 12.8T + 289T^{2} \) |
| 19 | \( 1 + (-2.63 + 2.63i)T - 361iT^{2} \) |
| 23 | \( 1 - 16.3T + 529T^{2} \) |
| 29 | \( 1 + (26.0 + 26.0i)T + 841iT^{2} \) |
| 31 | \( 1 - 20.2iT - 961T^{2} \) |
| 37 | \( 1 + (41.2 - 41.2i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 - 3.29iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (0.786 + 0.786i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 - 79.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-1.06 + 1.06i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + (32.5 + 32.5i)T + 3.48e3iT^{2} \) |
| 61 | \( 1 + (15.2 + 15.2i)T + 3.72e3iT^{2} \) |
| 67 | \( 1 + (60.0 - 60.0i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 - 56.3T + 5.04e3T^{2} \) |
| 73 | \( 1 - 9.70iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 84.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (26.7 - 26.7i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 - 115. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 146.T + 9.40e3T^{2} \) |
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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
−9.703295576016063679519748608806, −9.359027908742283327165160862738, −8.156052755819975005049865575721, −7.37338027671999969950543843549, −6.65231006469999583384282488081, −5.59924099785520371380328792569, −4.82635575658603751992454383208, −3.57838372346465538282961829297, −2.78720011658679223310657015287, −1.30971585975397005246660388110,
0.24586260173011279235043628546, 1.80046093408871849594241342659, 3.08262745336625497548507604340, 3.94927002705153903519685234139, 5.06462178873551776911914956996, 5.98640472822587813968829562688, 6.87626499233774983718254721123, 7.52635332443795249596097798022, 8.698492982885657171732421914743, 9.299407868172990717369171712077