L(s) = 1 | + (−0.589 + 0.808i)2-s + (0.991 − 0.127i)3-s + (−0.305 − 0.952i)4-s + (−0.709 + 0.704i)5-s + (−0.481 + 0.876i)6-s + (0.997 + 0.0637i)7-s + (0.949 + 0.313i)8-s + (0.967 − 0.252i)9-s + (−0.150 − 0.988i)10-s + (−0.939 − 0.343i)11-s + (−0.424 − 0.905i)12-s + (−0.663 − 0.748i)13-s + (−0.639 + 0.768i)14-s + (−0.614 + 0.788i)15-s + (−0.812 + 0.582i)16-s + (0.993 − 0.111i)17-s + ⋯ |
L(s) = 1 | + (−0.589 + 0.808i)2-s + (0.991 − 0.127i)3-s + (−0.305 − 0.952i)4-s + (−0.709 + 0.704i)5-s + (−0.481 + 0.876i)6-s + (0.997 + 0.0637i)7-s + (0.949 + 0.313i)8-s + (0.967 − 0.252i)9-s + (−0.150 − 0.988i)10-s + (−0.939 − 0.343i)11-s + (−0.424 − 0.905i)12-s + (−0.663 − 0.748i)13-s + (−0.639 + 0.768i)14-s + (−0.614 + 0.788i)15-s + (−0.812 + 0.582i)16-s + (0.993 − 0.111i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.935 - 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.935 - 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.404622282 - 0.2564761114i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.404622282 - 0.2564761114i\) |
\(L(1)\) |
\(\approx\) |
\(0.9760118570 + 0.2165882698i\) |
\(L(1)\) |
\(\approx\) |
\(0.9760118570 + 0.2165882698i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 4729 | \( 1 \) |
good | 2 | \( 1 + (-0.589 + 0.808i)T \) |
| 3 | \( 1 + (0.991 - 0.127i)T \) |
| 5 | \( 1 + (-0.709 + 0.704i)T \) |
| 7 | \( 1 + (0.997 + 0.0637i)T \) |
| 11 | \( 1 + (-0.939 - 0.343i)T \) |
| 13 | \( 1 + (-0.663 - 0.748i)T \) |
| 17 | \( 1 + (0.993 - 0.111i)T \) |
| 19 | \( 1 + (-0.996 - 0.0796i)T \) |
| 23 | \( 1 + (0.793 - 0.608i)T \) |
| 29 | \( 1 + (0.589 + 0.808i)T \) |
| 31 | \( 1 + (-0.872 + 0.488i)T \) |
| 37 | \( 1 + (-0.933 - 0.358i)T \) |
| 41 | \( 1 + (0.848 + 0.529i)T \) |
| 43 | \( 1 + (-0.198 - 0.980i)T \) |
| 47 | \( 1 + (-0.321 - 0.947i)T \) |
| 53 | \( 1 + (-0.821 + 0.569i)T \) |
| 59 | \( 1 + (0.410 + 0.912i)T \) |
| 61 | \( 1 + (0.395 + 0.918i)T \) |
| 67 | \( 1 + (0.721 - 0.692i)T \) |
| 71 | \( 1 + (0.495 - 0.868i)T \) |
| 73 | \( 1 + (-0.773 + 0.633i)T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + (0.627 - 0.778i)T \) |
| 89 | \( 1 + (-0.166 + 0.986i)T \) |
| 97 | \( 1 + (0.997 + 0.0637i)T \) |
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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
−18.55042223797519032662422342695, −17.39950323185428915812745768839, −17.15478099678020484001524086752, −16.12738159964478628992622215615, −15.66188704100023115038322853156, −14.68755148483514734762246421482, −14.2936669669859710714346231685, −13.208942206145715338153576465919, −12.80816704986248296802553095230, −12.12860526682754704649886561726, −11.37817307322382221054748225765, −10.7428544225215279785558758290, −9.89321092634933072146163914620, −9.348787052266539627248924225295, −8.5980470757515978049275702660, −7.946074004298801009304031677, −7.72647264276827090674530834479, −6.93188202991098321939247859961, −5.20907665828603953051779583074, −4.67359396663449536949314807514, −4.067684644655379146372648239906, −3.31138841544164792684982838807, −2.366317094240372615266680798235, −1.79916977513986124799912653745, −0.98046981597547496226907572877,
0.45630525172248728994441970841, 1.55214340043213645532205904247, 2.48475451314386872813300623855, 3.12930509220651844375644348865, 4.15879998275757549169158738563, 4.96108656622858995637538186113, 5.58055327625209976390083550378, 6.79038851470428499201679297350, 7.311840177242476986902784579448, 7.829408168189144970919769536891, 8.43297536005860925170749896292, 8.83140086465043397441953795387, 9.997771014109081441881381046754, 10.58259785630710885751504668168, 10.92166004963560702529100355429, 12.17352302052358025065978390538, 12.8192629100262627538814285667, 13.78467109047079334556380567865, 14.4288376723555928524836699617, 14.8529382354736488120724551531, 15.24502672014082820939986669323, 15.99477052281748150064506045810, 16.679576994385972419800499230215, 17.687177224531019617984607459451, 18.14423312903966696107764089124