L(s) = 1 | + (−0.888 − 0.458i)2-s + (−0.5 + 0.866i)3-s + (0.580 + 0.814i)4-s + (0.928 − 0.371i)5-s + (0.841 − 0.540i)6-s + (−0.142 − 0.989i)8-s + (−0.5 − 0.866i)9-s + (−0.995 − 0.0950i)10-s + (−0.995 + 0.0950i)12-s + (0.415 − 0.909i)13-s + (−0.142 + 0.989i)15-s + (−0.327 + 0.945i)16-s + (0.723 − 0.690i)17-s + (0.0475 + 0.998i)18-s + (0.723 + 0.690i)19-s + (0.841 + 0.540i)20-s + ⋯ |
L(s) = 1 | + (−0.888 − 0.458i)2-s + (−0.5 + 0.866i)3-s + (0.580 + 0.814i)4-s + (0.928 − 0.371i)5-s + (0.841 − 0.540i)6-s + (−0.142 − 0.989i)8-s + (−0.5 − 0.866i)9-s + (−0.995 − 0.0950i)10-s + (−0.995 + 0.0950i)12-s + (0.415 − 0.909i)13-s + (−0.142 + 0.989i)15-s + (−0.327 + 0.945i)16-s + (0.723 − 0.690i)17-s + (0.0475 + 0.998i)18-s + (0.723 + 0.690i)19-s + (0.841 + 0.540i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.740 - 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.740 - 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8625651779 - 0.3331530805i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8625651779 - 0.3331530805i\) |
\(L(1)\) |
\(\approx\) |
\(0.7447870427 - 0.08011837159i\) |
\(L(1)\) |
\(\approx\) |
\(0.7447870427 - 0.08011837159i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.888 - 0.458i)T \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 + (0.928 - 0.371i)T \) |
| 13 | \( 1 + (0.415 - 0.909i)T \) |
| 17 | \( 1 + (0.723 - 0.690i)T \) |
| 19 | \( 1 + (0.723 + 0.690i)T \) |
| 23 | \( 1 + (-0.327 + 0.945i)T \) |
| 29 | \( 1 + (-0.959 + 0.281i)T \) |
| 31 | \( 1 + (-0.995 - 0.0950i)T \) |
| 37 | \( 1 + (0.580 - 0.814i)T \) |
| 41 | \( 1 + (0.841 - 0.540i)T \) |
| 43 | \( 1 + (-0.142 - 0.989i)T \) |
| 47 | \( 1 + (0.0475 - 0.998i)T \) |
| 53 | \( 1 + (-0.327 - 0.945i)T \) |
| 59 | \( 1 + (-0.888 + 0.458i)T \) |
| 61 | \( 1 + (0.0475 - 0.998i)T \) |
| 67 | \( 1 + (0.0475 + 0.998i)T \) |
| 71 | \( 1 + (-0.959 + 0.281i)T \) |
| 73 | \( 1 + (0.981 - 0.189i)T \) |
| 79 | \( 1 + (0.928 - 0.371i)T \) |
| 83 | \( 1 + (-0.654 + 0.755i)T \) |
| 89 | \( 1 + (0.235 + 0.971i)T \) |
| 97 | \( 1 + (-0.142 - 0.989i)T \) |
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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
−22.3154293052459176800171810658, −21.41630995817571368379853985771, −20.41940401085715744105993117077, −19.502910199867222665523718430405, −18.58862362320739543131393742206, −18.34408971160711754058708305906, −17.471174819721899814103198899036, −16.75938066779874074431217651633, −16.22781739551455531684349105554, −14.86082625397396210010000073386, −14.19489314317598137296605270259, −13.417092022765742500030663079106, −12.40966066648670179606078955836, −11.28878630772514075543917319652, −10.817110409606455002930958898788, −9.743442193115504296593922452887, −9.02741169223448489520833447338, −7.93130029167643326567634710408, −7.19385014021242266737123374972, −6.240353525247635531695154672729, −5.93305980974459257394371846017, −4.78432294327252277244633236628, −2.874240409989989211879198125623, −1.873233365355167952070182329609, −1.14519358947925571028875247898,
0.6867925276171343468514248297, 1.82448449884483146874476766915, 3.1269097065712427019136315915, 3.872242205890038706059442013985, 5.42412747983991792446395690296, 5.74728812393805808729782850794, 7.12228831855556125988996319128, 8.12960741975833303983669930467, 9.23993750201771369350748190680, 9.61875633725862892870532118247, 10.41943653641863527665071772130, 11.1342928324591966528733882783, 12.07460095802250851661539031841, 12.82608536450511347441077933352, 13.88858367757822864077252955748, 14.978158537637469154692386340692, 16.02960446033609408384617707532, 16.4719636754152190377569395122, 17.28139878740337926318845869012, 17.9999895887360943020965259306, 18.50116294903897008599869414368, 19.89963019237648247420042916607, 20.57249494062607971487641155337, 20.98396207739853014176634222873, 21.89045981559618702758031016598