L(s) = 1 | + (0.965 − 0.258i)2-s + (−1.73 − 0.0795i)3-s + (0.866 − 0.499i)4-s + (−1.69 + 0.370i)6-s + (0.622 + 2.32i)7-s + (0.707 − 0.707i)8-s + (2.98 + 0.275i)9-s + (0.991 + 0.572i)11-s + (−1.53 + 0.796i)12-s + (0.640 − 2.38i)13-s + (1.20 + 2.08i)14-s + (0.500 − 0.866i)16-s + (4.99 + 4.99i)17-s + (2.95 − 0.507i)18-s − 2.78i·19-s + ⋯ |
L(s) = 1 | + (0.683 − 0.183i)2-s + (−0.998 − 0.0459i)3-s + (0.433 − 0.249i)4-s + (−0.690 + 0.151i)6-s + (0.235 + 0.877i)7-s + (0.249 − 0.249i)8-s + (0.995 + 0.0917i)9-s + (0.299 + 0.172i)11-s + (−0.444 + 0.229i)12-s + (0.177 − 0.662i)13-s + (0.321 + 0.556i)14-s + (0.125 − 0.216i)16-s + (1.21 + 1.21i)17-s + (0.696 − 0.119i)18-s − 0.638i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0253i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0253i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.67788 + 0.0212777i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.67788 + 0.0212777i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.965 + 0.258i)T \) |
| 3 | \( 1 + (1.73 + 0.0795i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-0.622 - 2.32i)T + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (-0.991 - 0.572i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.640 + 2.38i)T + (-11.2 - 6.5i)T^{2} \) |
| 17 | \( 1 + (-4.99 - 4.99i)T + 17iT^{2} \) |
| 19 | \( 1 + 2.78iT - 19T^{2} \) |
| 23 | \( 1 + (-5.95 - 1.59i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (-0.672 + 1.16i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.25 - 2.16i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-8.16 + 8.16i)T - 37iT^{2} \) |
| 41 | \( 1 + (1.70 - 0.986i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (8.68 - 2.32i)T + (37.2 - 21.5i)T^{2} \) |
| 47 | \( 1 + (11.9 - 3.19i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (1.84 - 1.84i)T - 53iT^{2} \) |
| 59 | \( 1 + (-1.31 - 2.27i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.54 - 6.13i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.0545 + 0.0146i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 9.10iT - 71T^{2} \) |
| 73 | \( 1 + (7.82 + 7.82i)T + 73iT^{2} \) |
| 79 | \( 1 + (8.46 + 4.88i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.724 + 2.70i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + 4.87T + 89T^{2} \) |
| 97 | \( 1 + (2.08 + 7.79i)T + (-84.0 + 48.5i)T^{2} \) |
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
−11.29116529410345373257020419881, −10.47333068896326679203788312600, −9.555330826867220994921720369744, −8.269385777443015235486916705448, −7.14559452879319630800694914701, −6.05428974908783134368387729671, −5.46420325578138016396794133191, −4.48647027852101655305074700279, −3.11171308103910214530365103215, −1.43690244896761297025514824742,
1.22944591686365643750469560017, 3.34512540671351805477860933922, 4.50656200668620566172994348533, 5.23073721879318241300241779597, 6.41993621735143083409488357562, 7.07008228434245971450114952411, 8.044394571475975340391664260395, 9.580034658137359917185797653171, 10.35801001572794979422209413351, 11.43195283324765763192311426566