L(s) = 1 | + 2-s + 4-s + (2.18 − 3.78i)5-s + 8-s + (2.18 − 3.78i)10-s + (−0.686 − 1.18i)11-s + (−1 − 1.73i)13-s + 16-s + (0.686 − 1.18i)17-s + (−2.5 − 4.33i)19-s + (2.18 − 3.78i)20-s + (−0.686 − 1.18i)22-s + (−0.813 + 1.40i)23-s + (−7.05 − 12.2i)25-s + (−1 − 1.73i)26-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + (0.977 − 1.69i)5-s + 0.353·8-s + (0.691 − 1.19i)10-s + (−0.206 − 0.358i)11-s + (−0.277 − 0.480i)13-s + 0.250·16-s + (0.166 − 0.288i)17-s + (−0.573 − 0.993i)19-s + (0.488 − 0.846i)20-s + (−0.146 − 0.253i)22-s + (−0.169 + 0.293i)23-s + (−1.41 − 2.44i)25-s + (−0.196 − 0.339i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.334 + 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.334 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.060963117\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.060963117\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-2.18 + 3.78i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (0.686 + 1.18i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1 + 1.73i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.686 + 1.18i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.5 + 4.33i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.813 - 1.40i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.37 - 7.57i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.31 - 4.00i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.05 + 7.02i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + (4.37 - 7.57i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 - 10.1T + 59T^{2} \) |
| 61 | \( 1 - 3.11T + 61T^{2} \) |
| 67 | \( 1 + 2.11T + 67T^{2} \) |
| 71 | \( 1 - 7.11T + 71T^{2} \) |
| 73 | \( 1 + (6.05 - 10.4i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + 5.11T + 79T^{2} \) |
| 83 | \( 1 + (-8.74 + 15.1i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-7.37 - 12.7i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.05 + 7.02i)T + (-48.5 - 84.0i)T^{2} \) |
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
−8.765084089910049097406821482094, −7.929247608464201991028101475104, −6.98408524316736152807980444999, −5.97692388736682339388444940426, −5.39012109277976157242743930542, −4.88374947389448649423613455752, −4.03871349875158789621991271476, −2.79342185460692474843589073734, −1.83582728903719678773161315886, −0.72715680072186635043737100441,
1.91947390343428096462302538737, 2.40207584929851166415748235320, 3.42234928239996472665550013074, 4.20502679160136556720991990647, 5.38169671979976514759539887121, 6.14557249483597741275419242083, 6.53561939169680485318808688618, 7.38254509006689396262598519544, 8.041229797508431632320502738878, 9.383114223047722714006774884000