L(s) = 1 | + (0.866 − 0.5i)2-s + (0.5 − 0.866i)4-s + (0.866 + 0.5i)7-s − i·8-s + (0.5 + 0.866i)11-s + 14-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s + (0.5 − 0.866i)19-s + (0.866 + 0.5i)22-s + (−0.866 + 0.5i)23-s + (0.866 − 0.5i)28-s + (−0.5 − 0.866i)29-s + 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)7-s − i·8-s + (0.5 + 0.866i)11-s + 14-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s + (0.5 − 0.866i)19-s + (0.866 + 0.5i)22-s + (−0.866 + 0.5i)23-s + (0.866 − 0.5i)28-s + (−0.5 − 0.866i)29-s + 31-s + (−0.866 − 0.5i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.680 - 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.680 - 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.847427166 - 0.8049111981i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.847427166 - 0.8049111981i\) |
\(L(1)\) |
\(\approx\) |
\(1.670074004 - 0.5160807222i\) |
\(L(1)\) |
\(\approx\) |
\(1.670074004 - 0.5160807222i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 7 | \( 1 + (0.866 + 0.5i)T \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
| 17 | \( 1 + (-0.866 - 0.5i)T \) |
| 19 | \( 1 + (0.5 - 0.866i)T \) |
| 23 | \( 1 + (-0.866 + 0.5i)T \) |
| 29 | \( 1 + (-0.5 - 0.866i)T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + (-0.866 + 0.5i)T \) |
| 41 | \( 1 + (0.5 + 0.866i)T \) |
| 43 | \( 1 + (-0.866 - 0.5i)T \) |
| 47 | \( 1 + iT \) |
| 53 | \( 1 - iT \) |
| 59 | \( 1 + (-0.5 + 0.866i)T \) |
| 61 | \( 1 + (-0.5 + 0.866i)T \) |
| 67 | \( 1 + (-0.866 + 0.5i)T \) |
| 71 | \( 1 + (0.5 - 0.866i)T \) |
| 73 | \( 1 + iT \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 - iT \) |
| 89 | \( 1 + (-0.5 - 0.866i)T \) |
| 97 | \( 1 + (0.866 + 0.5i)T \) |
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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
−26.789314977221709394437401407264, −26.255447419213804526112000891942, −24.78172179217791392301870184849, −24.37025449877509557611263065541, −23.44435710968433638397571079448, −22.400940167498415768318675301391, −21.58483856269932529515891721572, −20.65076079803296229340895062478, −19.77968957342914750890585093843, −18.26841582832154352534866389136, −17.22010587727101183467281496359, −16.44152014421072811567305782721, −15.35967675764456680791427164176, −14.23930517980243750703974894920, −13.78585734500309379709392973757, −12.46248682501276238911640774111, −11.481003012937317871055346069100, −10.56379333771922540089726449899, −8.72115524261237620254068545936, −7.87658880276802879718778721684, −6.684270259603811435158903090587, −5.628829272969401842679235942231, −4.4248880421934798916453259452, −3.47149666979144673921126999425, −1.80899619790097582398272143542,
1.5954171398544974959560426946, 2.67123703967928618649418847369, 4.2563337053881008551967286987, 5.03472822446031094322483228151, 6.30437035958419461955433426142, 7.49319682478844020339172153974, 9.053852699020082618059155823209, 10.1119468887298005174690241369, 11.49433536066656426132819871732, 11.86689967136477097001004895843, 13.19948800864554583591205074001, 14.09817513510775893405468481531, 15.117321937197969248085552145335, 15.76226558215133051234970170067, 17.418614583222382344795505853, 18.27187678865469108060246801456, 19.538244969472286387272311312038, 20.33472931955517856179964586465, 21.21098714000030631804377396766, 22.1693618323211885411557815902, 22.85941859088928494685801415387, 24.14648203315985717255677477693, 24.59371561310857880219519690570, 25.72014932138863024000783093863, 27.12554415882072718791891122108