| L(s) = 1 | + (−1.22 − 1.22i)7-s + (4.89 − 4.89i)13-s − i·19-s − 11·31-s + (3.67 + 3.67i)37-s + (−1.22 + 1.22i)43-s − 4i·49-s − 13·61-s + (−2.44 − 2.44i)67-s + (11.0 − 11.0i)73-s − 17i·79-s − 11.9·91-s + (−13.4 − 13.4i)97-s + (11.0 − 11.0i)103-s − 17i·109-s + ⋯ |
| L(s) = 1 | + (−0.462 − 0.462i)7-s + (1.35 − 1.35i)13-s − 0.229i·19-s − 1.97·31-s + (0.604 + 0.604i)37-s + (−0.186 + 0.186i)43-s − 0.571i·49-s − 1.66·61-s + (−0.299 − 0.299i)67-s + (1.29 − 1.29i)73-s − 1.91i·79-s − 1.25·91-s + (−1.36 − 1.36i)97-s + (1.08 − 1.08i)103-s − 1.62i·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.326 + 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.326 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.280117508\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.280117508\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.22 + 1.22i)T + 7iT^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + (-4.89 + 4.89i)T - 13iT^{2} \) |
| 17 | \( 1 - 17iT^{2} \) |
| 19 | \( 1 + iT - 19T^{2} \) |
| 23 | \( 1 + 23iT^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 11T + 31T^{2} \) |
| 37 | \( 1 + (-3.67 - 3.67i)T + 37iT^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + (1.22 - 1.22i)T - 43iT^{2} \) |
| 47 | \( 1 - 47iT^{2} \) |
| 53 | \( 1 + 53iT^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 13T + 61T^{2} \) |
| 67 | \( 1 + (2.44 + 2.44i)T + 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (-11.0 + 11.0i)T - 73iT^{2} \) |
| 79 | \( 1 + 17iT - 79T^{2} \) |
| 83 | \( 1 + 83iT^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + (13.4 + 13.4i)T + 97iT^{2} \) |
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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
−8.595096026645056213640938772330, −7.84701317530430810669470701060, −7.15087900272592552772718447678, −6.20504498624315287506384758225, −5.66551046804745776812943739220, −4.65221667740561259942507988268, −3.59080181579912975356016583160, −3.10685622833679522461648293065, −1.65525077607382128959125455487, −0.42838917458464759543812164754,
1.36966559820013828753140596949, 2.37480333503231177609426619159, 3.57442077756952284761679525120, 4.13706127745492484979591573616, 5.30792150387826184499176451060, 6.07198893958214063053562280342, 6.67317741507777992249531624530, 7.52635295224057315026672874679, 8.446829327961886120857820456414, 9.171678192665862027493538355264
