L(s) = 1 | + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s − 0.332i·5-s + (−0.866 + 0.5i)7-s + 0.999i·8-s + (0.166 − 0.288i)10-s + (−2.26 − 1.30i)11-s + (3.41 + 1.16i)13-s − 0.999·14-s + (−0.5 + 0.866i)16-s + (1.94 + 3.36i)17-s + (4.85 − 2.80i)19-s + (0.288 − 0.166i)20-s + (−1.30 − 2.26i)22-s + (−2.10 + 3.64i)23-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s − 0.148i·5-s + (−0.327 + 0.188i)7-s + 0.353i·8-s + (0.0526 − 0.0911i)10-s + (−0.681 − 0.393i)11-s + (0.946 + 0.323i)13-s − 0.267·14-s + (−0.125 + 0.216i)16-s + (0.471 + 0.816i)17-s + (1.11 − 0.643i)19-s + (0.0644 − 0.0372i)20-s + (−0.278 − 0.481i)22-s + (−0.439 + 0.760i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.311 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.311 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.351053428\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.351053428\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.866 - 0.5i)T \) |
| 13 | \( 1 + (-3.41 - 1.16i)T \) |
good | 5 | \( 1 + 0.332iT - 5T^{2} \) |
| 11 | \( 1 + (2.26 + 1.30i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-1.94 - 3.36i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.85 + 2.80i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.10 - 3.64i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.593 - 1.02i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 7.07iT - 31T^{2} \) |
| 37 | \( 1 + (-0.499 - 0.288i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.451 + 0.260i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.53 - 2.66i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 12.0iT - 47T^{2} \) |
| 53 | \( 1 - 9.71T + 53T^{2} \) |
| 59 | \( 1 + (9.07 - 5.23i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.71 + 6.44i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-10.3 - 5.96i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.818 + 0.472i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 4.66iT - 73T^{2} \) |
| 79 | \( 1 + 0.943T + 79T^{2} \) |
| 83 | \( 1 + 13.7iT - 83T^{2} \) |
| 89 | \( 1 + (2.92 + 1.69i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (5.03 - 2.90i)T + (48.5 - 84.0i)T^{2} \) |
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
−9.377313244684780833146798927034, −8.668825064834070714053365438228, −7.88224074426394905918903894761, −7.07322499525216528596329389030, −6.14765154880555819688879936364, −5.53358998092128056145189061409, −4.66836609041800098213331958733, −3.54162698824933471898446459937, −2.89452496836120799673419166325, −1.33471838791788171415647508572,
0.819170151948762855511968623689, 2.30616552449167365414273615638, 3.23990902169170392037671910448, 4.04969479874292474326173540571, 5.15710476354905533571861542277, 5.77966528849822350646334928410, 6.74940014561860881513918390024, 7.54038928009900666139134357339, 8.367750895887518300530450217858, 9.467122341450245322681825679221