L(s) = 1 | + (−1.73 − 1.41i)5-s + 4.24·7-s − 2.44·11-s + 2.44i·13-s + 6.92·17-s − 6.92i·19-s − 6i·23-s + (0.999 + 4.89i)25-s + 2.82i·29-s − 3.46i·31-s + (−7.34 − 6i)35-s − 2.44i·37-s − 7.07i·41-s + 8.48·43-s + 10.9·49-s + ⋯ |
L(s) = 1 | + (−0.774 − 0.632i)5-s + 1.60·7-s − 0.738·11-s + 0.679i·13-s + 1.68·17-s − 1.58i·19-s − 1.25i·23-s + (0.199 + 0.979i)25-s + 0.525i·29-s − 0.622i·31-s + (−1.24 − 1.01i)35-s − 0.402i·37-s − 1.10i·41-s + 1.29·43-s + 1.57·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.713 + 0.700i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.713 + 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.41594 - 0.579217i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.41594 - 0.579217i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.73 + 1.41i)T \) |
good | 7 | \( 1 - 4.24T + 7T^{2} \) |
| 11 | \( 1 + 2.44T + 11T^{2} \) |
| 13 | \( 1 - 2.44iT - 13T^{2} \) |
| 17 | \( 1 - 6.92T + 17T^{2} \) |
| 19 | \( 1 + 6.92iT - 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 - 2.82iT - 29T^{2} \) |
| 31 | \( 1 + 3.46iT - 31T^{2} \) |
| 37 | \( 1 + 2.44iT - 37T^{2} \) |
| 41 | \( 1 + 7.07iT - 41T^{2} \) |
| 43 | \( 1 - 8.48T + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 12.2T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 9.79T + 71T^{2} \) |
| 73 | \( 1 - 4.89iT - 73T^{2} \) |
| 79 | \( 1 + 10.3iT - 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 - 7.07iT - 89T^{2} \) |
| 97 | \( 1 - 14.6iT - 97T^{2} \) |
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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
−10.54286190368475994602417604106, −9.229477420348540797391742453860, −8.484184992071730712745762699046, −7.78945291975814197820439106820, −7.10986133560682068234959926653, −5.48649658996916779040125924150, −4.85563751760718394826333902969, −4.01673308301056584107856853347, −2.44614347347559622644770360676, −0.943702108044158655222633330718,
1.41028965816569564585058153464, 2.96844952529009270716045107133, 3.97907840386382966471183928470, 5.16554673129161741867577590780, 5.87211588201092960360143365214, 7.54243871950764162015206694743, 7.76252186958412576674482279793, 8.415073462363140253559750962012, 9.983268225300008166188829191038, 10.48857650096148169164668540931