L(s) = 1 | − 1.41i·2-s − 2.00·4-s + (1.38 − 4.80i)5-s + (5.24 + 4.63i)7-s + 2.82i·8-s + (−6.79 − 1.95i)10-s − 11.7·11-s − 24.8·13-s + (6.54 − 7.42i)14-s + 4.00·16-s + 7.26·17-s + 23.0i·19-s + (−2.76 + 9.61i)20-s + 16.6i·22-s − 26.4i·23-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.500·4-s + (0.276 − 0.961i)5-s + (0.749 + 0.661i)7-s + 0.353i·8-s + (−0.679 − 0.195i)10-s − 1.06·11-s − 1.90·13-s + (0.467 − 0.530i)14-s + 0.250·16-s + 0.427·17-s + 1.21i·19-s + (−0.138 + 0.480i)20-s + 0.756i·22-s − 1.14i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.538 - 0.842i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.538 - 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1064082587\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1064082587\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.38 + 4.80i)T \) |
| 7 | \( 1 + (-5.24 - 4.63i)T \) |
good | 11 | \( 1 + 11.7T + 121T^{2} \) |
| 13 | \( 1 + 24.8T + 169T^{2} \) |
| 17 | \( 1 - 7.26T + 289T^{2} \) |
| 19 | \( 1 - 23.0iT - 361T^{2} \) |
| 23 | \( 1 + 26.4iT - 529T^{2} \) |
| 29 | \( 1 + 57.0T + 841T^{2} \) |
| 31 | \( 1 + 10.2iT - 961T^{2} \) |
| 37 | \( 1 - 14.2iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 16.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 82.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 51.6T + 2.20e3T^{2} \) |
| 53 | \( 1 - 64.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 81.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 13.1iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 22.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 91.5T + 5.04e3T^{2} \) |
| 73 | \( 1 - 71.9T + 5.32e3T^{2} \) |
| 79 | \( 1 - 45.2T + 6.24e3T^{2} \) |
| 83 | \( 1 - 17.7T + 6.88e3T^{2} \) |
| 89 | \( 1 - 77.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 6.15T + 9.40e3T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.886592167041076772024945430605, −9.081031451672629647869820674414, −8.151806393522566030380871754300, −7.52845552557601375545200614045, −5.66082875732554758975538447338, −5.17861911664296820767930200775, −4.26557910198386185024509488180, −2.61537166802845832221011325511, −1.77462718617690340761850611481, −0.03601137126565305912714320327,
2.12438751676064588725021154927, 3.37525320800774924137094206054, 4.84510290754739152951561279302, 5.39171008762737958251118419059, 6.74473018153685902895325001344, 7.57225090304013528105080593305, 7.78365089474425781247744347502, 9.411195061167962782004022824425, 9.947487994007255206321277951621, 10.91689561703882396535533056955