L(s) = 1 | + (1 + 1.73i)4-s + (1.5 + 2.59i)5-s + (−0.5 + 0.866i)7-s + (3 − 5.19i)11-s + (2 + 3.46i)13-s + (−1.99 + 3.46i)16-s − 3·17-s + 2·19-s + (−3 + 5.19i)20-s + (−3 − 5.19i)23-s + (−2 + 3.46i)25-s − 1.99·28-s + (−3 + 5.19i)29-s + (2 + 3.46i)31-s − 3·35-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)4-s + (0.670 + 1.16i)5-s + (−0.188 + 0.327i)7-s + (0.904 − 1.56i)11-s + (0.554 + 0.960i)13-s + (−0.499 + 0.866i)16-s − 0.727·17-s + 0.458·19-s + (−0.670 + 1.16i)20-s + (−0.625 − 1.08i)23-s + (−0.400 + 0.692i)25-s − 0.377·28-s + (−0.557 + 0.964i)29-s + (0.359 + 0.622i)31-s − 0.507·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.39431 + 1.16997i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.39431 + 1.16997i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
good | 2 | \( 1 + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-3 + 5.19i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2 - 3.46i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 7T + 37T^{2} \) |
| 41 | \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.5 + 7.79i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + (-4.5 - 7.79i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5 + 8.66i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.5 + 2.59i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + (-5 + 8.66i)T + (-48.5 - 84.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.06534654478471713218347486185, −10.26979916749910016244597506302, −8.870280280708048604130048125561, −8.582199632440406613455595974165, −6.95568505175323002066440911478, −6.63902207319802501609197302043, −5.76300241416461614138720331550, −3.91536476538112132226318090407, −3.13157565604207640057341576461, −2.03285933843507897008001771059,
1.13684157457683013998255757108, 2.11314492646761721516340161378, 4.03437824152938906022421265140, 5.09063330831019722904981543595, 5.87257436771050111079399301616, 6.81453495192950695903546593208, 7.82511348005378655248352941424, 9.152193956593355017179658185189, 9.656478578336960530803698753126, 10.31635335272397056064590550797