L(s) = 1 | + (−1.89 − 3.29i)5-s + (0.841 − 2.50i)7-s + (−2.25 + 3.90i)11-s + (0.588 − 1.01i)13-s + (2.95 + 5.12i)17-s + (−2.55 + 4.42i)19-s + (2.09 + 3.62i)23-s + (−4.71 + 8.17i)25-s + (−2.11 − 3.65i)29-s + 6.24·31-s + (−9.85 + 1.99i)35-s + (−3.87 + 6.70i)37-s + (−0.754 + 1.30i)41-s + (5.01 + 8.68i)43-s − 2.23·47-s + ⋯ |
L(s) = 1 | + (−0.849 − 1.47i)5-s + (0.318 − 0.948i)7-s + (−0.680 + 1.17i)11-s + (0.163 − 0.282i)13-s + (0.717 + 1.24i)17-s + (−0.586 + 1.01i)19-s + (0.435 + 0.755i)23-s + (−0.943 + 1.63i)25-s + (−0.392 − 0.679i)29-s + 1.12·31-s + (−1.66 + 0.337i)35-s + (−0.636 + 1.10i)37-s + (−0.117 + 0.204i)41-s + (0.765 + 1.32i)43-s − 0.326·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 - 0.455i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.890 - 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.169862602\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.169862602\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.841 + 2.50i)T \) |
good | 5 | \( 1 + (1.89 + 3.29i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (2.25 - 3.90i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.588 + 1.01i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.95 - 5.12i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.55 - 4.42i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.09 - 3.62i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.11 + 3.65i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 6.24T + 31T^{2} \) |
| 37 | \( 1 + (3.87 - 6.70i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.754 - 1.30i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.01 - 8.68i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 2.23T + 47T^{2} \) |
| 53 | \( 1 + (6.49 + 11.2i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 12.3T + 59T^{2} \) |
| 61 | \( 1 - 1.45T + 61T^{2} \) |
| 67 | \( 1 + 1.62T + 67T^{2} \) |
| 71 | \( 1 - 8.48T + 71T^{2} \) |
| 73 | \( 1 + (-3.72 - 6.46i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 1.84T + 79T^{2} \) |
| 83 | \( 1 + (0.307 + 0.532i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.25 + 2.17i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.36 - 4.10i)T + (-48.5 + 84.0i)T^{2} \) |
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
−8.348881116614174778552773196184, −8.136486617530177030445853768969, −7.60979021904604192754618812550, −6.59387469180685487876292071802, −5.49122397870650086248426613147, −4.79128241699858954568549785639, −4.16845630859806376778817041413, −3.48808723832897656986943441968, −1.81116505435379924143944624921, −0.992866727924142115121689944754,
0.44664320054469224391619433393, 2.45012728281936564420824733352, 2.87268733763798103488448030679, 3.70764036006880862063224658461, 4.86584051230399438640361224042, 5.64113112142430933918159772005, 6.52341524922827218642071463946, 7.15736048742315457817016441116, 7.87682044667142738557073437029, 8.625548501005726406690833588613