L(s) = 1 | + (−1.23 − 2.13i)2-s + (−2.02 + 3.51i)4-s + 2.59·5-s + 5.05·8-s + (−3.19 − 5.52i)10-s − 4.51·11-s + (−0.5 − 0.866i)13-s + (−2.16 − 3.74i)16-s + (−0.472 − 0.819i)17-s + (2.02 − 3.51i)19-s + (−5.25 + 9.10i)20-s + (5.55 + 9.61i)22-s + 0.273·23-s + 1.72·25-s + (−1.23 + 2.13i)26-s + ⋯ |
L(s) = 1 | + (−0.869 − 1.50i)2-s + (−1.01 + 1.75i)4-s + 1.15·5-s + 1.78·8-s + (−1.00 − 1.74i)10-s − 1.36·11-s + (−0.138 − 0.240i)13-s + (−0.540 − 0.936i)16-s + (−0.114 − 0.198i)17-s + (0.465 − 0.805i)19-s + (−1.17 + 2.03i)20-s + (1.18 + 2.05i)22-s + 0.0569·23-s + 0.345·25-s + (−0.241 + 0.417i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 - 0.252i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.967 - 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7298628577\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7298628577\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (1.23 + 2.13i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 - 2.59T + 5T^{2} \) |
| 11 | \( 1 + 4.51T + 11T^{2} \) |
| 13 | \( 1 + (0.5 + 0.866i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (0.472 + 0.819i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.02 + 3.51i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 0.273T + 23T^{2} \) |
| 29 | \( 1 + (-1.23 + 2.13i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (1.16 - 2.01i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.890 - 1.54i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3.20 + 5.54i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.21 + 9.03i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (6.08 + 10.5i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.13 + 5.43i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (1.36 - 2.36i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.13 - 1.96i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.90 + 13.6i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 3.27T + 71T^{2} \) |
| 73 | \( 1 + (-0.753 - 1.30i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (7.35 + 12.7i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.472 - 0.819i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (7.17 - 12.4i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.74 + 9.95i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.425484555679529219652347656352, −8.738595614057078191547725517438, −7.924977700697490733089894260075, −6.94565904049979821272251518974, −5.63092439835807257128732629063, −4.88527795041514598952790044638, −3.45475130251187506488412073808, −2.54597394948905562684377700139, −1.88828143425112024942326679699, −0.40322055123403983971888824935,
1.42787887609306606369133999650, 2.79662716044767984207115529881, 4.62158125647881367813605239187, 5.51831059777579691564750979528, 5.97301978494337878250047827147, 6.82195327481966481281092210724, 7.75191349096464853731443439711, 8.228589446347102888491465106054, 9.255348137580683749413638740509, 9.792453882151679188418962778801