L(s) = 1 | + (1.20 + 0.736i)2-s + (0.914 + 1.77i)4-s + 1.08i·5-s + (−0.207 + 2.82i)8-s + (−0.797 + 1.30i)10-s + 2.08i·11-s + 2.61i·13-s + (−2.32 + 3.25i)16-s − 4.46i·17-s − 1.12·19-s + (−1.92 + 0.989i)20-s + (−1.53 + 2.51i)22-s + 7.11i·23-s + 3.82·25-s + (−1.92 + 3.15i)26-s + ⋯ |
L(s) = 1 | + (0.853 + 0.521i)2-s + (0.457 + 0.889i)4-s + 0.484i·5-s + (−0.0732 + 0.997i)8-s + (−0.252 + 0.413i)10-s + 0.628i·11-s + 0.724i·13-s + (−0.582 + 0.813i)16-s − 1.08i·17-s − 0.258·19-s + (−0.430 + 0.221i)20-s + (−0.327 + 0.536i)22-s + 1.48i·23-s + 0.765·25-s + (−0.377 + 0.618i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.780 - 0.624i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.780 - 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.496956445\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.496956445\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.20 - 0.736i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 1.08iT - 5T^{2} \) |
| 11 | \( 1 - 2.08iT - 11T^{2} \) |
| 13 | \( 1 - 2.61iT - 13T^{2} \) |
| 17 | \( 1 + 4.46iT - 17T^{2} \) |
| 19 | \( 1 + 1.12T + 19T^{2} \) |
| 23 | \( 1 - 7.11iT - 23T^{2} \) |
| 29 | \( 1 - 1.17T + 29T^{2} \) |
| 31 | \( 1 + 7.70T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + 5.54iT - 41T^{2} \) |
| 43 | \( 1 - 7.97iT - 43T^{2} \) |
| 47 | \( 1 - 5.44T + 47T^{2} \) |
| 53 | \( 1 - 6.48T + 53T^{2} \) |
| 59 | \( 1 + 8.82T + 59T^{2} \) |
| 61 | \( 1 - 13.0iT - 61T^{2} \) |
| 67 | \( 1 + 10.0iT - 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 7.97iT - 73T^{2} \) |
| 79 | \( 1 - 4.16iT - 79T^{2} \) |
| 83 | \( 1 - 4.31T + 83T^{2} \) |
| 89 | \( 1 - 4.01iT - 89T^{2} \) |
| 97 | \( 1 - 3.82iT - 97T^{2} \) |
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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
−9.400905341930517296740188666672, −8.844903837795277742719985682860, −7.54168044228543023987660427482, −7.25369868437197948223324846121, −6.45940339815753232088175858480, −5.50255968079596157509742654567, −4.75853754911695563826367509836, −3.83627355501274882655575733580, −2.93381265723579301954906721011, −1.88255628763068660203725018144,
0.67469276887324311426059706139, 1.95363382364094375570057042803, 3.07369086066665307742814317111, 3.93246217242978761109293089812, 4.82429440723239203213961808552, 5.63145996920713806617923685455, 6.31238968845515714084955142399, 7.23702513866643936883497666439, 8.412290206913716251824154403797, 8.900594852773484442860458183211