L(s) = 1 | − 0.585·5-s + 0.828i·7-s − 2.82i·11-s − 2.82i·13-s + 2.58i·17-s + 5.65·19-s − 6.82·23-s − 4.65·25-s + 3.41·29-s − 8.82i·31-s − 0.485i·35-s + 7.65i·37-s − 5.41i·41-s − 1.65·43-s − 4.48·47-s + ⋯ |
L(s) = 1 | − 0.261·5-s + 0.313i·7-s − 0.852i·11-s − 0.784i·13-s + 0.627i·17-s + 1.29·19-s − 1.42·23-s − 0.931·25-s + 0.634·29-s − 1.58i·31-s − 0.0820i·35-s + 1.25i·37-s − 0.845i·41-s − 0.252·43-s − 0.654·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.169 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.122124225\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.122124225\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 0.585T + 5T^{2} \) |
| 7 | \( 1 - 0.828iT - 7T^{2} \) |
| 11 | \( 1 + 2.82iT - 11T^{2} \) |
| 13 | \( 1 + 2.82iT - 13T^{2} \) |
| 17 | \( 1 - 2.58iT - 17T^{2} \) |
| 19 | \( 1 - 5.65T + 19T^{2} \) |
| 23 | \( 1 + 6.82T + 23T^{2} \) |
| 29 | \( 1 - 3.41T + 29T^{2} \) |
| 31 | \( 1 + 8.82iT - 31T^{2} \) |
| 37 | \( 1 - 7.65iT - 37T^{2} \) |
| 41 | \( 1 + 5.41iT - 41T^{2} \) |
| 43 | \( 1 + 1.65T + 43T^{2} \) |
| 47 | \( 1 + 4.48T + 47T^{2} \) |
| 53 | \( 1 + 9.07T + 53T^{2} \) |
| 59 | \( 1 + 13.6iT - 59T^{2} \) |
| 61 | \( 1 + 3.65iT - 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 + 10.4iT - 79T^{2} \) |
| 83 | \( 1 + 10.8iT - 83T^{2} \) |
| 89 | \( 1 + 3.75iT - 89T^{2} \) |
| 97 | \( 1 - 2.34T + 97T^{2} \) |
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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
−8.599491167129431158384700039420, −8.050271216412762769725683686380, −7.48043908972701649949554990412, −6.17421436061508557572265711817, −5.84415165181649602377775222850, −4.83674971239023945928439012182, −3.75497182336446420327204412285, −3.07963901199321753829676663192, −1.85139346523114944866609755625, −0.39731439156352676821313039217,
1.31052383753195925082816742878, 2.43709650691966385980552313137, 3.59470178482629253270225822431, 4.38320070169824173031072658436, 5.17000955006923245417223006111, 6.15550643422303230488989934119, 7.09185158304741850142025012164, 7.52988058826543269913042939204, 8.405517681459513328654948241159, 9.354759205045159633551393369804