L(s) = 1 | + (−4.45 − 0.786i)3-s + (5.56 + 4.66i)5-s + (4.24 + 7.35i)7-s + (10.8 + 3.93i)9-s + (−4.58 + 7.94i)11-s + (8.90 − 1.57i)13-s + (−21.1 − 25.1i)15-s + (−6.76 + 2.46i)17-s + (−18.6 + 3.58i)19-s + (−13.1 − 36.1i)21-s + (28.1 − 23.6i)23-s + (4.81 + 27.2i)25-s + (−9.83 − 5.68i)27-s + (−16.9 + 46.6i)29-s + (30.9 − 17.8i)31-s + ⋯ |
L(s) = 1 | + (−1.48 − 0.262i)3-s + (1.11 + 0.933i)5-s + (0.606 + 1.05i)7-s + (1.20 + 0.437i)9-s + (−0.416 + 0.722i)11-s + (0.685 − 0.120i)13-s + (−1.40 − 1.67i)15-s + (−0.398 + 0.144i)17-s + (−0.982 + 0.188i)19-s + (−0.626 − 1.72i)21-s + (1.22 − 1.02i)23-s + (0.192 + 1.09i)25-s + (−0.364 − 0.210i)27-s + (−0.585 + 1.60i)29-s + (0.998 − 0.576i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.533 - 0.845i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.533 - 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.825906 + 0.455586i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.825906 + 0.455586i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (18.6 - 3.58i)T \) |
good | 3 | \( 1 + (4.45 + 0.786i)T + (8.45 + 3.07i)T^{2} \) |
| 5 | \( 1 + (-5.56 - 4.66i)T + (4.34 + 24.6i)T^{2} \) |
| 7 | \( 1 + (-4.24 - 7.35i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (4.58 - 7.94i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-8.90 + 1.57i)T + (158. - 57.8i)T^{2} \) |
| 17 | \( 1 + (6.76 - 2.46i)T + (221. - 185. i)T^{2} \) |
| 23 | \( 1 + (-28.1 + 23.6i)T + (91.8 - 520. i)T^{2} \) |
| 29 | \( 1 + (16.9 - 46.6i)T + (-644. - 540. i)T^{2} \) |
| 31 | \( 1 + (-30.9 + 17.8i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + 61.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (21.8 + 3.84i)T + (1.57e3 + 574. i)T^{2} \) |
| 43 | \( 1 + (50.2 + 42.1i)T + (321. + 1.82e3i)T^{2} \) |
| 47 | \( 1 + (-67.7 - 24.6i)T + (1.69e3 + 1.41e3i)T^{2} \) |
| 53 | \( 1 + (-5.17 - 6.17i)T + (-487. + 2.76e3i)T^{2} \) |
| 59 | \( 1 + (-3.10 - 8.52i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-32.9 + 27.6i)T + (646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (-2.86 + 7.87i)T + (-3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (-5.02 + 5.98i)T + (-875. - 4.96e3i)T^{2} \) |
| 73 | \( 1 + (-17.9 + 102. i)T + (-5.00e3 - 1.82e3i)T^{2} \) |
| 79 | \( 1 + (5.92 + 1.04i)T + (5.86e3 + 2.13e3i)T^{2} \) |
| 83 | \( 1 + (-17.5 - 30.3i)T + (-3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-56.6 + 9.98i)T + (7.44e3 - 2.70e3i)T^{2} \) |
| 97 | \( 1 + (36.6 + 100. i)T + (-7.20e3 + 6.04e3i)T^{2} \) |
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
−14.55208704382947821825962771377, −13.09136002434296922383027603453, −12.22530770382460453999384934835, −10.93725837506199745617427002525, −10.50079124001112527871940885850, −8.861706376889644080306487480685, −6.91966342464125029280309835139, −6.02438257804819083007278602180, −5.07669734740961281984975595770, −2.14977654994249382426844519535,
1.06612530648745692039017289826, 4.52820432842057797772687280617, 5.49304850298228743609249697831, 6.57874233990168421133384445747, 8.440190803942820821779617186523, 9.903627366782263099966356396514, 10.85863288548680798405885363703, 11.64308723578401219549339121477, 13.24407849921751030241274869891, 13.57802066310895441059170803437