L(s) = 1 | + (1.30 − 0.541i)2-s + (−0.541 − 1.64i)3-s + (1.41 − 1.41i)4-s + (1.25 + 1.84i)5-s + (−1.59 − 1.85i)6-s − 3.29·7-s + (1.08 − 2.61i)8-s + (−2.41 + 1.78i)9-s + (2.64 + 1.73i)10-s + 2.51i·11-s + (−3.09 − 1.56i)12-s + 4.65·13-s + (−4.29 + 1.78i)14-s + (2.35 − 3.07i)15-s − 4i·16-s − 3.69·17-s + ⋯ |
L(s) = 1 | + (0.923 − 0.382i)2-s + (−0.312 − 0.949i)3-s + (0.707 − 0.707i)4-s + (0.563 + 0.826i)5-s + (−0.652 − 0.758i)6-s − 1.24·7-s + (0.382 − 0.923i)8-s + (−0.804 + 0.593i)9-s + (0.836 + 0.547i)10-s + 0.759i·11-s + (−0.892 − 0.450i)12-s + 1.29·13-s + (−1.14 + 0.475i)14-s + (0.609 − 0.793i)15-s − i·16-s − 0.896·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.499 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.499 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.33994 - 0.773886i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.33994 - 0.773886i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.30 + 0.541i)T \) |
| 3 | \( 1 + (0.541 + 1.64i)T \) |
| 5 | \( 1 + (-1.25 - 1.84i)T \) |
good | 7 | \( 1 + 3.29T + 7T^{2} \) |
| 11 | \( 1 - 2.51iT - 11T^{2} \) |
| 13 | \( 1 - 4.65T + 13T^{2} \) |
| 17 | \( 1 + 3.69T + 17T^{2} \) |
| 19 | \( 1 - 0.828T + 19T^{2} \) |
| 23 | \( 1 - 2.61iT - 23T^{2} \) |
| 29 | \( 1 + 6.08T + 29T^{2} \) |
| 31 | \( 1 + 1.17iT - 31T^{2} \) |
| 37 | \( 1 - 1.92T + 37T^{2} \) |
| 41 | \( 1 + 8.59iT - 41T^{2} \) |
| 43 | \( 1 + 6.01iT - 43T^{2} \) |
| 47 | \( 1 + 2.61iT - 47T^{2} \) |
| 53 | \( 1 - 4.59iT - 53T^{2} \) |
| 59 | \( 1 + 2.51iT - 59T^{2} \) |
| 61 | \( 1 + 8.48iT - 61T^{2} \) |
| 67 | \( 1 - 3.29iT - 67T^{2} \) |
| 71 | \( 1 + 7.12T + 71T^{2} \) |
| 73 | \( 1 + 6.58iT - 73T^{2} \) |
| 79 | \( 1 - 16.4iT - 79T^{2} \) |
| 83 | \( 1 - 9.37T + 83T^{2} \) |
| 89 | \( 1 - 5.03iT - 89T^{2} \) |
| 97 | \( 1 - 2.72iT - 97T^{2} \) |
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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
−13.33761635717045763598720842897, −12.55052807930329813822125317494, −11.36937843464211649078378111476, −10.55051007138240382507141372839, −9.339017263736030275639144478923, −7.23209975231332345957134162247, −6.48188703773792509111590339537, −5.65830789690905855248975594480, −3.54179762518627065481637119395, −2.10943270936933276531839013596,
3.22249956800353534867804659320, 4.42741160417538150296028343842, 5.81387765548919653004957860550, 6.37104874028319363874590334363, 8.476278288091162100926259460114, 9.364532668483405125757916021815, 10.69188950686124340811532934847, 11.67061278243001089490208566455, 13.01764294211238761946548685894, 13.41394666130375005076358764493