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
−9.274570396488311879559731476423, −8.761193429203823549023786381158, −8.111333658770422673046411932608, −7.52271778582452305793769971840, −6.07850896175115489122246205855, −5.59527103437783591402794362513, −4.43207301239049410792395046731, −3.85519067543223914671907053787, −2.37861122301919103211179971729, −1.24089748236615196252704064180,
0.74966894341745383791670731190, 2.17635141672523722364374449014, 3.56649661964306406201555707919, 4.29197538013761702960063477262, 4.96934050589377282884348060566, 6.43192587984488291840272553204, 7.21127016424453969779578445485, 7.53211892055321177047510509693, 8.717955188352665728061041937757, 9.197771582267832621863671176668