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
−13.41394666130375005076358764493, −13.01764294211238761946548685894, −11.67061278243001089490208566455, −10.69188950686124340811532934847, −9.364532668483405125757916021815, −8.476278288091162100926259460114, −6.37104874028319363874590334363, −5.81387765548919653004957860550, −4.42741160417538150296028343842, −3.22249956800353534867804659320,
2.10943270936933276531839013596, 3.54179762518627065481637119395, 5.65830789690905855248975594480, 6.48188703773792509111590339537, 7.23209975231332345957134162247, 9.339017263736030275639144478923, 10.55051007138240382507141372839, 11.36937843464211649078378111476, 12.55052807930329813822125317494, 13.33761635717045763598720842897