L(s) = 1 | + (0.866 + 0.5i)2-s + (0.5 + 0.866i)4-s + (0.866 + 0.5i)5-s + i·8-s + (0.5 + 0.866i)10-s + (0.866 − 0.5i)11-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (−0.866 − 0.5i)19-s + i·20-s + 22-s + (−0.5 + 0.866i)23-s + (0.5 + 0.866i)25-s − 29-s + (0.866 − 0.5i)31-s + (−0.866 + 0.5i)32-s + ⋯ |
L(s) = 1 | + (0.866 + 0.5i)2-s + (0.5 + 0.866i)4-s + (0.866 + 0.5i)5-s + i·8-s + (0.5 + 0.866i)10-s + (0.866 − 0.5i)11-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (−0.866 − 0.5i)19-s + i·20-s + 22-s + (−0.5 + 0.866i)23-s + (0.5 + 0.866i)25-s − 29-s + (0.866 − 0.5i)31-s + (−0.866 + 0.5i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.349 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.349 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.898030195 + 1.317295258i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.898030195 + 1.317295258i\) |
\(L(1)\) |
\(\approx\) |
\(1.718921942 + 0.7569514352i\) |
\(L(1)\) |
\(\approx\) |
\(1.718921942 + 0.7569514352i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 + (0.866 + 0.5i)T \) |
| 11 | \( 1 + (0.866 - 0.5i)T \) |
| 17 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 + (-0.866 - 0.5i)T \) |
| 23 | \( 1 + (-0.5 + 0.866i)T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 + (0.866 - 0.5i)T \) |
| 37 | \( 1 + (0.866 + 0.5i)T \) |
| 41 | \( 1 - iT \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + (-0.866 - 0.5i)T \) |
| 53 | \( 1 + (0.5 + 0.866i)T \) |
| 59 | \( 1 + (0.866 - 0.5i)T \) |
| 61 | \( 1 + (-0.5 + 0.866i)T \) |
| 67 | \( 1 + (0.866 - 0.5i)T \) |
| 71 | \( 1 - iT \) |
| 73 | \( 1 + (-0.866 + 0.5i)T \) |
| 79 | \( 1 + (-0.5 + 0.866i)T \) |
| 83 | \( 1 - iT \) |
| 89 | \( 1 + (-0.866 - 0.5i)T \) |
| 97 | \( 1 + iT \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−25.15929577659941670158077186026, −24.63945901852297801364352180874, −23.66384238374497181642996398101, −22.64914351897255866638569922265, −21.84512256893405610963135685573, −21.11436864056333468604979821610, −20.21445271835393410526363949035, −19.47322247435149370802082633636, −18.26512336659239815369669729514, −17.165655276987720880873107489547, −16.28868046864084652870402278093, −14.90807209651236337507769459089, −14.36596392464286858292501617751, −13.15949590118382373158030216137, −12.632321456603661489722909384993, −11.55800998283664896910639922310, −10.38409416953632651163662814742, −9.61969747522446328116479722078, −8.4415575152061840633210008116, −6.64651441161728679079050484816, −6.023433925372479159635151544283, −4.761715378040968313197858678066, −3.91581484910928015210052074736, −2.32555232006560521208377276346, −1.44815680842568669196518181619,
1.96223697104925034890612309862, 3.088409274252249262447154656200, 4.28489618161643468522049171341, 5.53707206997284632578239447032, 6.39415248053309884703327714406, 7.19666914184150368575570591791, 8.58494244674683300563381480293, 9.65858729171633752112231778210, 11.06117382358985424906276855102, 11.79073126046544048830562922507, 13.199401829271052854336583554261, 13.72079018379754602788880681554, 14.65471046304922140208226252619, 15.47800651309281819808603188359, 16.702714564985387829120435804089, 17.34790615088346621855655675605, 18.36422673125357417717716596564, 19.64341744088417422567679471163, 20.717874666194723125453596802916, 21.67893603859083278274066073828, 22.18296565159319623604921052358, 23.062101943205842737729939233908, 24.18997364710749574652796349579, 24.89507906088240984610549672591, 25.7094905529654810775386008991