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
−21.89045981559618702758031016598, −20.98396207739853014176634222873, −20.57249494062607971487641155337, −19.89963019237648247420042916607, −18.50116294903897008599869414368, −17.9999895887360943020965259306, −17.28139878740337926318845869012, −16.4719636754152190377569395122, −16.02960446033609408384617707532, −14.978158537637469154692386340692, −13.88858367757822864077252955748, −12.82608536450511347441077933352, −12.07460095802250851661539031841, −11.1342928324591966528733882783, −10.41943653641863527665071772130, −9.61875633725862892870532118247, −9.23993750201771369350748190680, −8.12960741975833303983669930467, −7.12228831855556125988996319128, −5.74728812393805808729782850794, −5.42412747983991792446395690296, −3.872242205890038706059442013985, −3.1269097065712427019136315915, −1.82448449884483146874476766915, −0.6867925276171343468514248297,
1.14519358947925571028875247898, 1.873233365355167952070182329609, 2.874240409989989211879198125623, 4.78432294327252277244633236628, 5.93305980974459257394371846017, 6.240353525247635531695154672729, 7.19385014021242266737123374972, 7.93130029167643326567634710408, 9.02741169223448489520833447338, 9.743442193115504296593922452887, 10.817110409606455002930958898788, 11.28878630772514075543917319652, 12.40966066648670179606078955836, 13.417092022765742500030663079106, 14.19489314317598137296605270259, 14.86082625397396210010000073386, 16.22781739551455531684349105554, 16.75938066779874074431217651633, 17.471174819721899814103198899036, 18.34408971160711754058708305906, 18.58862362320739543131393742206, 19.502910199867222665523718430405, 20.41940401085715744105993117077, 21.41630995817571368379853985771, 22.3154293052459176800171810658