L(s) = 1 | + (−1 + i)2-s − 2i·4-s + (−3 − 4i)5-s + (8 − 8i)7-s + (2 + 2i)8-s + (7 + i)10-s + 4·11-s + (−3 − 3i)13-s + 16i·14-s − 4·16-s + (19 − 19i)17-s + 8i·19-s + (−8 + 6i)20-s + (−4 + 4i)22-s + (−20 − 20i)23-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.5i)2-s − 0.5i·4-s + (−0.600 − 0.800i)5-s + (1.14 − 1.14i)7-s + (0.250 + 0.250i)8-s + (0.700 + 0.100i)10-s + 0.363·11-s + (−0.230 − 0.230i)13-s + 1.14i·14-s − 0.250·16-s + (1.11 − 1.11i)17-s + 0.421i·19-s + (−0.400 + 0.300i)20-s + (−0.181 + 0.181i)22-s + (−0.869 − 0.869i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.767 + 0.640i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.767 + 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.929943 - 0.337071i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.929943 - 0.337071i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 - i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (3 + 4i)T \) |
good | 7 | \( 1 + (-8 + 8i)T - 49iT^{2} \) |
| 11 | \( 1 - 4T + 121T^{2} \) |
| 13 | \( 1 + (3 + 3i)T + 169iT^{2} \) |
| 17 | \( 1 + (-19 + 19i)T - 289iT^{2} \) |
| 19 | \( 1 - 8iT - 361T^{2} \) |
| 23 | \( 1 + (20 + 20i)T + 529iT^{2} \) |
| 29 | \( 1 - 38iT - 841T^{2} \) |
| 31 | \( 1 + 44T + 961T^{2} \) |
| 37 | \( 1 + (3 - 3i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 - 70T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-36 - 36i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-17 - 17i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 - 92iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 72T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-44 + 44i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 + 88T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-55 - 55i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 + 12iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (24 + 24i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + 26iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (57 - 57i)T - 9.40e3iT^{2} \) |
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
−14.15886357832638319280918651839, −12.62924959685238902986063294303, −11.49901903319005434993482542731, −10.44556285674008257262123730504, −9.138106399083541669950966414132, −7.923884175503687284138153359726, −7.32213427269648630663092833128, −5.35773864414447254213450436352, −4.17151599413298949152826555678, −1.04637085856452170468959364729,
2.13577502529342843796375824345, 3.89414961913487455801271052203, 5.77026034946687879387660380905, 7.52441484965829760354957143396, 8.362558544002381284914929252314, 9.639434373450386469354397017371, 10.96430840121788125065986836627, 11.67597479320255105685364595619, 12.47636677792100653882590862659, 14.23602651858445477500073167637