L(s) = 1 | + (0.707 + 0.707i)2-s + (0.751 + 1.56i)3-s + 1.00i·4-s + (−2.21 + 0.330i)5-s + (−0.571 + 1.63i)6-s + (−0.310 + 0.310i)7-s + (−0.707 + 0.707i)8-s + (−1.86 + 2.34i)9-s + (−1.79 − 1.32i)10-s + i·11-s + (−1.56 + 0.751i)12-s + (0.758 + 0.758i)13-s − 0.438·14-s + (−2.17 − 3.20i)15-s − 1.00·16-s + (1.97 + 1.97i)17-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + (0.434 + 0.900i)3-s + 0.500i·4-s + (−0.988 + 0.147i)5-s + (−0.233 + 0.667i)6-s + (−0.117 + 0.117i)7-s + (−0.250 + 0.250i)8-s + (−0.623 + 0.782i)9-s + (−0.568 − 0.420i)10-s + 0.301i·11-s + (−0.450 + 0.217i)12-s + (0.210 + 0.210i)13-s − 0.117·14-s + (−0.562 − 0.826i)15-s − 0.250·16-s + (0.478 + 0.478i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.737 - 0.675i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.737 - 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.555274 + 1.42866i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.555274 + 1.42866i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 3 | \( 1 + (-0.751 - 1.56i)T \) |
| 5 | \( 1 + (2.21 - 0.330i)T \) |
| 11 | \( 1 - iT \) |
good | 7 | \( 1 + (0.310 - 0.310i)T - 7iT^{2} \) |
| 13 | \( 1 + (-0.758 - 0.758i)T + 13iT^{2} \) |
| 17 | \( 1 + (-1.97 - 1.97i)T + 17iT^{2} \) |
| 19 | \( 1 - 1.37iT - 19T^{2} \) |
| 23 | \( 1 + (-2.73 + 2.73i)T - 23iT^{2} \) |
| 29 | \( 1 - 1.28T + 29T^{2} \) |
| 31 | \( 1 - 3.91T + 31T^{2} \) |
| 37 | \( 1 + (-5.63 + 5.63i)T - 37iT^{2} \) |
| 41 | \( 1 - 4.53iT - 41T^{2} \) |
| 43 | \( 1 + (6.86 + 6.86i)T + 43iT^{2} \) |
| 47 | \( 1 + (-7.19 - 7.19i)T + 47iT^{2} \) |
| 53 | \( 1 + (-4.23 + 4.23i)T - 53iT^{2} \) |
| 59 | \( 1 + 1.01T + 59T^{2} \) |
| 61 | \( 1 - 6.22T + 61T^{2} \) |
| 67 | \( 1 + (7.31 - 7.31i)T - 67iT^{2} \) |
| 71 | \( 1 - 3.60iT - 71T^{2} \) |
| 73 | \( 1 + (11.0 + 11.0i)T + 73iT^{2} \) |
| 79 | \( 1 - 12.3iT - 79T^{2} \) |
| 83 | \( 1 + (-9.76 + 9.76i)T - 83iT^{2} \) |
| 89 | \( 1 - 9.69T + 89T^{2} \) |
| 97 | \( 1 + (-2.59 + 2.59i)T - 97iT^{2} \) |
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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
−11.95165878546516639755170835883, −11.03859534821431149275269871840, −10.12103474396767721731683969392, −8.927346105977420914432180178116, −8.166804655515965917931269242435, −7.26093681055202836911246971326, −5.97081534123102032879152559508, −4.70252939776093494023759116804, −3.93112021196059697711300194142, −2.85447794357114358704714735937,
0.947387403783473197675048880488, 2.80375504359123440548150251944, 3.72751885185476863395903013548, 5.11693316512293279574525553747, 6.44522028068003269149194555735, 7.41910263569506954060141276571, 8.312203929813254172200543087851, 9.261828593223492397670682405925, 10.54422946567551811656197861476, 11.71853827566386830101932003027