L(s) = 1 | + (0.951 − 0.309i)2-s + (−0.587 + 0.809i)3-s + (0.809 − 0.587i)4-s + (−0.309 + 0.951i)6-s + (0.587 + 0.809i)7-s + (0.587 − 0.809i)8-s + (−0.309 − 0.951i)9-s + i·12-s + (−0.951 + 0.309i)13-s + (0.809 + 0.587i)14-s + (0.309 − 0.951i)16-s + (−0.951 − 0.309i)17-s + (−0.587 − 0.809i)18-s + (−0.809 − 0.587i)19-s − 21-s + ⋯ |
L(s) = 1 | + (0.951 − 0.309i)2-s + (−0.587 + 0.809i)3-s + (0.809 − 0.587i)4-s + (−0.309 + 0.951i)6-s + (0.587 + 0.809i)7-s + (0.587 − 0.809i)8-s + (−0.309 − 0.951i)9-s + i·12-s + (−0.951 + 0.309i)13-s + (0.809 + 0.587i)14-s + (0.309 − 0.951i)16-s + (−0.951 − 0.309i)17-s + (−0.587 − 0.809i)18-s + (−0.809 − 0.587i)19-s − 21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.247352780 + 0.1062066600i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.247352780 + 0.1062066600i\) |
\(L(1)\) |
\(\approx\) |
\(1.384027223 + 0.06411751476i\) |
\(L(1)\) |
\(\approx\) |
\(1.384027223 + 0.06411751476i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.951 - 0.309i)T \) |
| 3 | \( 1 + (-0.587 + 0.809i)T \) |
| 7 | \( 1 + (0.587 + 0.809i)T \) |
| 13 | \( 1 + (-0.951 + 0.309i)T \) |
| 17 | \( 1 + (-0.951 - 0.309i)T \) |
| 19 | \( 1 + (-0.809 - 0.587i)T \) |
| 23 | \( 1 - iT \) |
| 29 | \( 1 + (-0.809 + 0.587i)T \) |
| 31 | \( 1 + (0.309 + 0.951i)T \) |
| 37 | \( 1 + (-0.587 - 0.809i)T \) |
| 41 | \( 1 + (0.809 + 0.587i)T \) |
| 43 | \( 1 + iT \) |
| 47 | \( 1 + (0.587 - 0.809i)T \) |
| 53 | \( 1 + (0.951 - 0.309i)T \) |
| 59 | \( 1 + (0.809 - 0.587i)T \) |
| 61 | \( 1 + (-0.309 + 0.951i)T \) |
| 67 | \( 1 + iT \) |
| 71 | \( 1 + (0.309 - 0.951i)T \) |
| 73 | \( 1 + (-0.587 - 0.809i)T \) |
| 79 | \( 1 + (0.309 + 0.951i)T \) |
| 83 | \( 1 + (0.951 + 0.309i)T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + (-0.951 + 0.309i)T \) |
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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
−33.369691919099007219888527893468, −31.890315105698219145555395506386, −30.773048063325668023510806408624, −29.83804565163015442224492475876, −29.15284212087660361390661699101, −27.51338259755597056317384996786, −26.002520115029621886196469517330, −24.63298958390636410199750162619, −23.98346832303542200586630662808, −22.96617571104121034521339677970, −21.96618227848747642377234965571, −20.5389857939140680017374093762, −19.32769577986783592592925561924, −17.47410042040384150988551225586, −16.92456298348977877228650014870, −15.22862535290310598521983933982, −13.91479321622556160090030789300, −12.954730059179764571783288361879, −11.74832315592209633624343508384, −10.65802019420099425487110199039, −7.96719660406043990749351994067, −7.01980041451009374580179115890, −5.6404744869139935069803192426, −4.26981074670714828746090483678, −2.07366721334378857674144358529,
2.46838343379528889907541660924, 4.36456043783324990822419228310, 5.281864042887035461200680907486, 6.70818183117028235697318067231, 9.059823396217572868533787355645, 10.59836497973071322411584132896, 11.59507781308331844231805470245, 12.64207897334552310793388840469, 14.47642066511400246301301528974, 15.25453856577886381565558045646, 16.4499819458379640027018815239, 17.927605482249229570164755834998, 19.60606160295174319655537087408, 20.93005829832175522925431843971, 21.76793831691038202869501814261, 22.54963864488306443129317487711, 23.87696378801827405158563867195, 24.832174890122814435788114485470, 26.51482659267726019914667266942, 27.87899466598695726792410582451, 28.65666410982032812564902096240, 29.78443828353706864957922104306, 31.19435442750361610112262516913, 31.97423204837713617473263986924, 33.11474880017254542783164963920