| L(s) = 1 | + (0.671 − 0.488i)2-s + (0.913 − 0.406i)3-s + (−0.405 + 1.24i)4-s + (0.5 + 0.866i)5-s + (0.415 − 0.719i)6-s + (0.743 + 0.825i)7-s + (0.849 + 2.61i)8-s + (0.669 − 0.743i)9-s + (0.758 + 0.337i)10-s + (−2.72 + 0.579i)11-s + (0.137 + 1.30i)12-s + (−0.521 + 4.95i)13-s + (0.902 + 0.191i)14-s + (0.809 + 0.587i)15-s + (−0.274 − 0.199i)16-s + (7.11 + 1.51i)17-s + ⋯ |
| L(s) = 1 | + (0.474 − 0.345i)2-s + (0.527 − 0.234i)3-s + (−0.202 + 0.623i)4-s + (0.223 + 0.387i)5-s + (0.169 − 0.293i)6-s + (0.280 + 0.312i)7-s + (0.300 + 0.924i)8-s + (0.223 − 0.247i)9-s + (0.239 + 0.106i)10-s + (−0.822 + 0.174i)11-s + (0.0395 + 0.376i)12-s + (−0.144 + 1.37i)13-s + (0.241 + 0.0512i)14-s + (0.208 + 0.151i)15-s + (−0.0686 − 0.0498i)16-s + (1.72 + 0.366i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.842 - 0.539i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.842 - 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.00305 + 0.586222i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.00305 + 0.586222i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.913 + 0.406i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 31 | \( 1 + (1.18 - 5.43i)T \) |
| good | 2 | \( 1 + (-0.671 + 0.488i)T + (0.618 - 1.90i)T^{2} \) |
| 7 | \( 1 + (-0.743 - 0.825i)T + (-0.731 + 6.96i)T^{2} \) |
| 11 | \( 1 + (2.72 - 0.579i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (0.521 - 4.95i)T + (-12.7 - 2.70i)T^{2} \) |
| 17 | \( 1 + (-7.11 - 1.51i)T + (15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 + (0.483 + 4.59i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (2.11 + 6.51i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-2.44 + 1.77i)T + (8.96 - 27.5i)T^{2} \) |
| 37 | \( 1 + (-3.90 + 6.77i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-7.99 - 3.55i)T + (27.4 + 30.4i)T^{2} \) |
| 43 | \( 1 + (-0.758 - 7.22i)T + (-42.0 + 8.94i)T^{2} \) |
| 47 | \( 1 + (8.43 + 6.12i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-7.32 + 8.13i)T + (-5.54 - 52.7i)T^{2} \) |
| 59 | \( 1 + (-7.59 + 3.38i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 + (3.61 + 6.25i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.99 + 3.32i)T + (-7.42 - 70.6i)T^{2} \) |
| 73 | \( 1 + (12.0 - 2.56i)T + (66.6 - 29.6i)T^{2} \) |
| 79 | \( 1 + (1.05 + 0.223i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (-8.63 - 3.84i)T + (55.5 + 61.6i)T^{2} \) |
| 89 | \( 1 + (2.01 - 6.20i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-2.08 + 6.40i)T + (-78.4 - 57.0i)T^{2} \) |
| 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
−11.32067102718218967741541051718, −10.29036191188568781290503462017, −9.255019987292914620131325721465, −8.332658102005138014822416299465, −7.60993046654567240077412330402, −6.56408644042970513499567851342, −5.18068326512404423418144445175, −4.19694981899824416287562823629, −2.97014860698432635671316510291, −2.11554347829636289670567480774,
1.20097748596519823884571899957, 3.07046795277811831792300404508, 4.25312054746384389356406714912, 5.48812729708255828092943901368, 5.74911484790437261404860815929, 7.59610189594227342811891960398, 7.936003980089114889779130861331, 9.342573208669956886990304955401, 10.12193155263156964007796649004, 10.55756380789090064219813914207
