L(s) = 1 | + 3-s + 5-s + (−2.18 + 3.78i)7-s + 9-s + (1.49 − 2.59i)11-s + (−1.04 − 1.80i)13-s + 15-s + (−0.454 − 0.786i)17-s + (3.53 + 6.11i)19-s + (−2.18 + 3.78i)21-s + (3.49 + 6.04i)23-s + 25-s + 27-s + (3.93 − 6.82i)29-s + (−3.83 + 6.65i)31-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + (−0.826 + 1.43i)7-s + 0.333·9-s + (0.450 − 0.780i)11-s + (−0.288 − 0.500i)13-s + 0.258·15-s + (−0.110 − 0.190i)17-s + (0.810 + 1.40i)19-s + (−0.477 + 0.826i)21-s + (0.727 + 1.26i)23-s + 0.200·25-s + 0.192·27-s + (0.731 − 1.26i)29-s + (−0.689 + 1.19i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.108 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.108 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.152150734\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.152150734\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 + (-2.50 - 7.79i)T \) |
good | 7 | \( 1 + (2.18 - 3.78i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.49 + 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.04 + 1.80i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (0.454 + 0.786i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.53 - 6.11i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.49 - 6.04i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.93 + 6.82i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (3.83 - 6.65i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.417 + 0.722i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.33 - 2.31i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + 12.0T + 43T^{2} \) |
| 47 | \( 1 + (-0.436 + 0.756i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 7.39T + 53T^{2} \) |
| 59 | \( 1 + 1.81T + 59T^{2} \) |
| 61 | \( 1 + (-1.99 - 3.44i)T + (-30.5 + 52.8i)T^{2} \) |
| 71 | \( 1 + (-0.0217 + 0.0377i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.49 - 7.79i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.453 - 0.785i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.41 - 2.44i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 2.76T + 89T^{2} \) |
| 97 | \( 1 + (-0.274 - 0.474i)T + (-48.5 + 84.0i)T^{2} \) |
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
−8.635878464772782579802216883379, −8.114544011882567381410246315698, −7.11581554101472096069553975846, −6.33419905895069952548049427081, −5.61233274721075178311783274932, −5.16966389647050672216152911671, −3.65709859391361156031792120217, −3.17071366567568691298612461218, −2.36695395109719928590992982189, −1.27558716054032251530298465531,
0.58485768703785367439546181915, 1.75032321520069711950298439218, 2.81772245999128301890967551670, 3.58905955231832658414730210144, 4.47037448616399456057842986679, 5.02394274178995325513401553689, 6.42977131548190960255483552347, 6.97915203911114138878017216160, 7.21348279614056248016105157197, 8.331131116130286482993456555991