L(s) = 1 | + (−1.48 − 1.48i)2-s + 2.43i·4-s + (−1.28 − 1.82i)5-s + (−1.75 − 1.97i)7-s + (0.640 − 0.640i)8-s + (−0.798 + 4.63i)10-s + 2.67·11-s + (−1.22 − 1.22i)13-s + (−0.320 + 5.55i)14-s + 2.95·16-s + (−4.74 + 4.74i)17-s − 6.01·19-s + (4.43 − 3.13i)20-s + (−3.97 − 3.97i)22-s + (0.175 − 0.175i)23-s + ⋯ |
L(s) = 1 | + (−1.05 − 1.05i)2-s + 1.21i·4-s + (−0.576 − 0.816i)5-s + (−0.665 − 0.746i)7-s + (0.226 − 0.226i)8-s + (−0.252 + 1.46i)10-s + 0.805·11-s + (−0.340 − 0.340i)13-s + (−0.0857 + 1.48i)14-s + 0.738·16-s + (−1.15 + 1.15i)17-s − 1.38·19-s + (0.992 − 0.701i)20-s + (−0.847 − 0.847i)22-s + (0.0366 − 0.0366i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.319 - 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.319 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0906497 + 0.126261i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0906497 + 0.126261i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (1.28 + 1.82i)T \) |
| 7 | \( 1 + (1.75 + 1.97i)T \) |
good | 2 | \( 1 + (1.48 + 1.48i)T + 2iT^{2} \) |
| 11 | \( 1 - 2.67T + 11T^{2} \) |
| 13 | \( 1 + (1.22 + 1.22i)T + 13iT^{2} \) |
| 17 | \( 1 + (4.74 - 4.74i)T - 17iT^{2} \) |
| 19 | \( 1 + 6.01T + 19T^{2} \) |
| 23 | \( 1 + (-0.175 + 0.175i)T - 23iT^{2} \) |
| 29 | \( 1 - 0.304iT - 29T^{2} \) |
| 31 | \( 1 - 7.25iT - 31T^{2} \) |
| 37 | \( 1 + (0.735 + 0.735i)T + 37iT^{2} \) |
| 41 | \( 1 + 7.05iT - 41T^{2} \) |
| 43 | \( 1 + (-0.304 + 0.304i)T - 43iT^{2} \) |
| 47 | \( 1 + (-0.556 + 0.556i)T - 47iT^{2} \) |
| 53 | \( 1 + (-4.99 + 4.99i)T - 53iT^{2} \) |
| 59 | \( 1 + 7.98T + 59T^{2} \) |
| 61 | \( 1 + 5.53iT - 61T^{2} \) |
| 67 | \( 1 + (3.43 + 3.43i)T + 67iT^{2} \) |
| 71 | \( 1 + 15.3T + 71T^{2} \) |
| 73 | \( 1 + (10.0 + 10.0i)T + 73iT^{2} \) |
| 79 | \( 1 + 11.2iT - 79T^{2} \) |
| 83 | \( 1 + (4.88 + 4.88i)T + 83iT^{2} \) |
| 89 | \( 1 - 6.91T + 89T^{2} \) |
| 97 | \( 1 + (-8.84 + 8.84i)T - 97iT^{2} \) |
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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
−10.74530324723892496348402647892, −10.28833935965682812291704122806, −8.970335139728034901278118581390, −8.709298593890885052771568519605, −7.47617635329672668923516710549, −6.25270456489123727094695345862, −4.45156015744022969151924584676, −3.47584597135220387846630630926, −1.72619440852232772257218134776, −0.15343485634099092428156353098,
2.64842818666299614528705020991, 4.24745615308348683931514593083, 6.05775346583352346527678635069, 6.67492173240778080701745989999, 7.42807218120521122030078947146, 8.604050498168495594562896343830, 9.235948788894909364217002313726, 10.10602001249284956413571629378, 11.29430559336038292569771457665, 12.06911219465640536944374765334