L(s) = 1 | + (0.847 − 1.46i)2-s + (0.0521 + 1.73i)3-s + (−0.435 − 0.753i)4-s + 0.0882·5-s + (2.58 + 1.39i)6-s + (1.84 + 3.19i)7-s + 1.91·8-s + (−2.99 + 0.180i)9-s + (0.0747 − 0.129i)10-s + (−1.97 − 3.42i)11-s + (1.28 − 0.792i)12-s + (−2.03 − 3.51i)13-s + 6.25·14-s + (0.00459 + 0.152i)15-s + (2.49 − 4.31i)16-s + (−0.586 − 1.01i)17-s + ⋯ |
L(s) = 1 | + (0.599 − 1.03i)2-s + (0.0300 + 0.999i)3-s + (−0.217 − 0.376i)4-s + 0.0394·5-s + (1.05 + 0.567i)6-s + (0.698 + 1.20i)7-s + 0.676·8-s + (−0.998 + 0.0601i)9-s + (0.0236 − 0.0409i)10-s + (−0.596 − 1.03i)11-s + (0.370 − 0.228i)12-s + (−0.563 − 0.975i)13-s + 1.67·14-s + (0.00118 + 0.0394i)15-s + (0.622 − 1.07i)16-s + (−0.142 − 0.246i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.178i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 + 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.62098 - 0.145853i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.62098 - 0.145853i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.0521 - 1.73i)T \) |
| 19 | \( 1 + (-3.26 - 2.89i)T \) |
good | 2 | \( 1 + (-0.847 + 1.46i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 - 0.0882T + 5T^{2} \) |
| 7 | \( 1 + (-1.84 - 3.19i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.97 + 3.42i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.03 + 3.51i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (0.586 + 1.01i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (1.91 + 3.31i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 6.56T + 29T^{2} \) |
| 31 | \( 1 + (4.14 - 7.18i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 6.88T + 37T^{2} \) |
| 41 | \( 1 + 4.66T + 41T^{2} \) |
| 43 | \( 1 + (-4.12 + 7.14i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 4.43T + 47T^{2} \) |
| 53 | \( 1 + (3.62 - 6.27i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 - 1.01T + 59T^{2} \) |
| 61 | \( 1 - 3.22T + 61T^{2} \) |
| 67 | \( 1 + (-1.45 - 2.51i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (4.36 + 7.56i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.43 - 5.95i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.65 + 4.59i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.34 - 5.80i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-4.41 + 7.64i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.894 - 1.55i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−12.30639903283845281193699316566, −11.86103168425767740904229612008, −10.76287886131075151872684071224, −10.19190570248418627799933447264, −8.794857161586055348629130287226, −7.925855685757095051574658217583, −5.60295707588436158565534183847, −5.01003944949459265852241669386, −3.46581589865515374309013830054, −2.48281781321006841407916297074,
1.82897085539709848336190663786, 4.27473517148119557203375674210, 5.32702391894132476551437879527, 6.71841107556673980313399045977, 7.37706156951767763780518644243, 7.988889774649226140055968846472, 9.741081658196884692165217005633, 10.98726064933599391406708700171, 12.01021720739460675153985674686, 13.24184006628563049377051346604