L(s) = 1 | + (−1.83 + 1.28i)5-s + (2.94 − 2.94i)7-s + 1.61·11-s + (2.50 + 2.50i)13-s + (−4.59 − 4.59i)17-s − 4i·19-s + (1.09 + 1.09i)23-s + (1.70 − 4.70i)25-s − 4.75·29-s + 5.01i·31-s + (−1.61 + 9.18i)35-s + (−2.50 + 2.50i)37-s + 9.18·41-s + (7.40 − 7.40i)43-s + (7.32 − 7.32i)47-s + ⋯ |
L(s) = 1 | + (−0.818 + 0.574i)5-s + (1.11 − 1.11i)7-s + 0.485·11-s + (0.696 + 0.696i)13-s + (−1.11 − 1.11i)17-s − 0.917i·19-s + (0.227 + 0.227i)23-s + (0.340 − 0.940i)25-s − 0.882·29-s + 0.901i·31-s + (−0.272 + 1.55i)35-s + (−0.412 + 0.412i)37-s + 1.43·41-s + (1.12 − 1.12i)43-s + (1.06 − 1.06i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.713 + 0.700i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.713 + 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.592116400\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.592116400\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.83 - 1.28i)T \) |
good | 7 | \( 1 + (-2.94 + 2.94i)T - 7iT^{2} \) |
| 11 | \( 1 - 1.61T + 11T^{2} \) |
| 13 | \( 1 + (-2.50 - 2.50i)T + 13iT^{2} \) |
| 17 | \( 1 + (4.59 + 4.59i)T + 17iT^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + (-1.09 - 1.09i)T + 23iT^{2} \) |
| 29 | \( 1 + 4.75T + 29T^{2} \) |
| 31 | \( 1 - 5.01iT - 31T^{2} \) |
| 37 | \( 1 + (2.50 - 2.50i)T - 37iT^{2} \) |
| 41 | \( 1 - 9.18T + 41T^{2} \) |
| 43 | \( 1 + (-7.40 + 7.40i)T - 43iT^{2} \) |
| 47 | \( 1 + (-7.32 + 7.32i)T - 47iT^{2} \) |
| 53 | \( 1 + (-3.11 - 3.11i)T + 53iT^{2} \) |
| 59 | \( 1 + 1.61iT - 59T^{2} \) |
| 61 | \( 1 + 6.78iT - 61T^{2} \) |
| 67 | \( 1 + (7.40 + 7.40i)T + 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (-5 + 5i)T - 73iT^{2} \) |
| 79 | \( 1 - 5.01T + 79T^{2} \) |
| 83 | \( 1 + (-7.57 + 7.57i)T - 83iT^{2} \) |
| 89 | \( 1 - 2.74iT - 89T^{2} \) |
| 97 | \( 1 + (-2.40 - 2.40i)T + 97iT^{2} \) |
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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
−9.197771582267832621863671176668, −8.717955188352665728061041937757, −7.53211892055321177047510509693, −7.21127016424453969779578445485, −6.43192587984488291840272553204, −4.96934050589377282884348060566, −4.29197538013761702960063477262, −3.56649661964306406201555707919, −2.17635141672523722364374449014, −0.74966894341745383791670731190,
1.24089748236615196252704064180, 2.37861122301919103211179971729, 3.85519067543223914671907053787, 4.43207301239049410792395046731, 5.59527103437783591402794362513, 6.07850896175115489122246205855, 7.52271778582452305793769971840, 8.111333658770422673046411932608, 8.761193429203823549023786381158, 9.274570396488311879559731476423