L(s) = 1 | + (−0.307 + 1.38i)2-s + 2.85i·3-s + (−1.81 − 0.849i)4-s + (1.43 − 1.71i)5-s + (−3.94 − 0.879i)6-s + (−0.458 + 0.458i)7-s + (1.73 − 2.23i)8-s − 5.15·9-s + (1.92 + 2.50i)10-s + (−0.492 + 0.492i)11-s + (2.42 − 5.17i)12-s + 4.52·13-s + (−0.492 − 0.774i)14-s + (4.89 + 4.09i)15-s + (2.55 + 3.07i)16-s + (−3.12 + 3.12i)17-s + ⋯ |
L(s) = 1 | + (−0.217 + 0.976i)2-s + 1.64i·3-s + (−0.905 − 0.424i)4-s + (0.641 − 0.766i)5-s + (−1.60 − 0.358i)6-s + (−0.173 + 0.173i)7-s + (0.611 − 0.791i)8-s − 1.71·9-s + (0.608 + 0.793i)10-s + (−0.148 + 0.148i)11-s + (0.700 − 1.49i)12-s + 1.25·13-s + (−0.131 − 0.207i)14-s + (1.26 + 1.05i)15-s + (0.638 + 0.769i)16-s + (−0.758 + 0.758i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.581 - 0.813i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.581 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.390367 + 0.759155i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.390367 + 0.759155i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.307 - 1.38i)T \) |
| 5 | \( 1 + (-1.43 + 1.71i)T \) |
good | 3 | \( 1 - 2.85iT - 3T^{2} \) |
| 7 | \( 1 + (0.458 - 0.458i)T - 7iT^{2} \) |
| 11 | \( 1 + (0.492 - 0.492i)T - 11iT^{2} \) |
| 13 | \( 1 - 4.52T + 13T^{2} \) |
| 17 | \( 1 + (3.12 - 3.12i)T - 17iT^{2} \) |
| 19 | \( 1 + (-4.04 + 4.04i)T - 19iT^{2} \) |
| 23 | \( 1 + (1.80 + 1.80i)T + 23iT^{2} \) |
| 29 | \( 1 + (3.83 + 3.83i)T + 29iT^{2} \) |
| 31 | \( 1 + 0.139iT - 31T^{2} \) |
| 37 | \( 1 - 5.84T + 37T^{2} \) |
| 41 | \( 1 + 4.55iT - 41T^{2} \) |
| 43 | \( 1 + 7.49T + 43T^{2} \) |
| 47 | \( 1 + (4.14 + 4.14i)T + 47iT^{2} \) |
| 53 | \( 1 - 2.75iT - 53T^{2} \) |
| 59 | \( 1 + (3.62 + 3.62i)T + 59iT^{2} \) |
| 61 | \( 1 + (-3.72 + 3.72i)T - 61iT^{2} \) |
| 67 | \( 1 - 3.32T + 67T^{2} \) |
| 71 | \( 1 - 1.37T + 71T^{2} \) |
| 73 | \( 1 + (2.55 - 2.55i)T - 73iT^{2} \) |
| 79 | \( 1 + 3.86T + 79T^{2} \) |
| 83 | \( 1 - 14.4iT - 83T^{2} \) |
| 89 | \( 1 + 3.35T + 89T^{2} \) |
| 97 | \( 1 + (4.95 - 4.95i)T - 97iT^{2} \) |
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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
−15.21899699083098878131627595399, −13.94029945099730860690337448875, −13.04569822918566939515944260488, −11.08777842613164331845478244036, −9.939674245250335959667958867035, −9.184930044296836798352740593310, −8.367256849495975304396262026000, −6.19860244182619703762897077051, −5.15757804600645716441508591873, −4.03832024927086418200604841773,
1.62712533632086963832074203688, 3.16493363326819554594006276673, 5.83792576967913157050149330710, 7.11668157904845414650908814513, 8.280073077024364940676002249149, 9.662315005026135290119300373323, 11.03291486367435442607709788175, 11.79204463255198116263366587072, 13.17373383567226504267781280765, 13.48817234163435190525674148711