L(s) = 1 | + (−0.224 − 0.224i)7-s − 2.78i·11-s + (1.22 − 1.22i)13-s + (−2.78 + 2.78i)17-s − 3.89i·19-s + (6.19 + 6.19i)23-s + 2.78·29-s + 0.449·31-s + (−3.67 − 3.67i)37-s − 9.60i·41-s + (−0.550 + 0.550i)43-s − 6.89i·49-s + (−6.19 − 6.19i)53-s − 12.3·59-s + 12.7·61-s + ⋯ |
L(s) = 1 | + (−0.0849 − 0.0849i)7-s − 0.839i·11-s + (0.339 − 0.339i)13-s + (−0.675 + 0.675i)17-s − 0.894i·19-s + (1.29 + 1.29i)23-s + 0.517·29-s + 0.0807·31-s + (−0.604 − 0.604i)37-s − 1.49i·41-s + (−0.0839 + 0.0839i)43-s − 0.985i·49-s + (−0.850 − 0.850i)53-s − 1.61·59-s + 1.63·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.502228428\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.502228428\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (0.224 + 0.224i)T + 7iT^{2} \) |
| 11 | \( 1 + 2.78iT - 11T^{2} \) |
| 13 | \( 1 + (-1.22 + 1.22i)T - 13iT^{2} \) |
| 17 | \( 1 + (2.78 - 2.78i)T - 17iT^{2} \) |
| 19 | \( 1 + 3.89iT - 19T^{2} \) |
| 23 | \( 1 + (-6.19 - 6.19i)T + 23iT^{2} \) |
| 29 | \( 1 - 2.78T + 29T^{2} \) |
| 31 | \( 1 - 0.449T + 31T^{2} \) |
| 37 | \( 1 + (3.67 + 3.67i)T + 37iT^{2} \) |
| 41 | \( 1 + 9.60iT - 41T^{2} \) |
| 43 | \( 1 + (0.550 - 0.550i)T - 43iT^{2} \) |
| 47 | \( 1 - 47iT^{2} \) |
| 53 | \( 1 + (6.19 + 6.19i)T + 53iT^{2} \) |
| 59 | \( 1 + 12.3T + 59T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 + (8.67 + 8.67i)T + 67iT^{2} \) |
| 71 | \( 1 - 9.60iT - 71T^{2} \) |
| 73 | \( 1 + (-6.22 + 6.22i)T - 73iT^{2} \) |
| 79 | \( 1 + 9.24iT - 79T^{2} \) |
| 83 | \( 1 + (3.41 + 3.41i)T + 83iT^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 + (-4.77 - 4.77i)T + 97iT^{2} \) |
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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
−8.824572295312348842867214410174, −7.964251485671897265432653940475, −7.10981622310273610762854698648, −6.42291457678862415073748944740, −5.55854114311543275915664520303, −4.84511587143782410638673931298, −3.71340079938248315335185866640, −3.07920973697673167607398371243, −1.83857484801837372801850260004, −0.53166910296946505787072621519,
1.19623874711284567220995489689, 2.37456406512619880650083476339, 3.25680228267707911762308643726, 4.49116939207625412022540665975, 4.84815566916419431916985113233, 6.09550445262821464905196672790, 6.68991507526392035434153959240, 7.42567963881215698155597600384, 8.315558390311524867038213332288, 8.994064035787211813259224484182