L(s) = 1 | + (−0.707 − 0.707i)2-s + (0.707 − 0.707i)3-s + i·4-s + (0.382 − 0.923i)5-s − 6-s + (0.382 + 0.923i)7-s + (0.707 − 0.707i)8-s − i·9-s + (−0.923 + 0.382i)10-s + (0.707 − 0.707i)11-s + (0.707 + 0.707i)12-s + (−0.382 + 0.923i)13-s + (0.382 − 0.923i)14-s + (−0.382 − 0.923i)15-s − 16-s + (−0.382 + 0.923i)17-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)2-s + (0.707 − 0.707i)3-s + i·4-s + (0.382 − 0.923i)5-s − 6-s + (0.382 + 0.923i)7-s + (0.707 − 0.707i)8-s − i·9-s + (−0.923 + 0.382i)10-s + (0.707 − 0.707i)11-s + (0.707 + 0.707i)12-s + (−0.382 + 0.923i)13-s + (0.382 − 0.923i)14-s + (−0.382 − 0.923i)15-s − 16-s + (−0.382 + 0.923i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0486 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0486 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6635714228 - 0.6966525242i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6635714228 - 0.6966525242i\) |
\(L(1)\) |
\(\approx\) |
\(0.8292624504 - 0.5304617907i\) |
\(L(1)\) |
\(\approx\) |
\(0.8292624504 - 0.5304617907i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 97 | \( 1 \) |
good | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (0.382 - 0.923i)T \) |
| 7 | \( 1 + (0.382 + 0.923i)T \) |
| 11 | \( 1 + (0.707 - 0.707i)T \) |
| 13 | \( 1 + (-0.382 + 0.923i)T \) |
| 17 | \( 1 + (-0.382 + 0.923i)T \) |
| 19 | \( 1 + (0.382 - 0.923i)T \) |
| 23 | \( 1 + (-0.923 - 0.382i)T \) |
| 29 | \( 1 + (-0.923 - 0.382i)T \) |
| 31 | \( 1 + (-0.707 + 0.707i)T \) |
| 37 | \( 1 + (0.923 - 0.382i)T \) |
| 41 | \( 1 + (0.923 + 0.382i)T \) |
| 43 | \( 1 + iT \) |
| 47 | \( 1 + iT \) |
| 53 | \( 1 + (0.707 + 0.707i)T \) |
| 59 | \( 1 + (-0.923 + 0.382i)T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 + (-0.382 + 0.923i)T \) |
| 71 | \( 1 + (0.923 - 0.382i)T \) |
| 73 | \( 1 - iT \) |
| 79 | \( 1 + (-0.707 + 0.707i)T \) |
| 83 | \( 1 + (0.382 - 0.923i)T \) |
| 89 | \( 1 + (0.707 - 0.707i)T \) |
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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
−30.28912443694643183419912464213, −29.36036750644000330700385790599, −27.637818687680838942685756234820, −27.17736489934507641243826333153, −26.22082981262359043762837255012, −25.43871712012455923740443959595, −24.542793782147726802592123386604, −22.94143832395573040500156432147, −22.202987429283951341343987228655, −20.434811608950944189587028804936, −19.93429397633467418410409920976, −18.509666923888102186386022538425, −17.52333832230022617729391634820, −16.47517316450189496952207731392, −15.15964467399975329404166496834, −14.478412085917524888864397045924, −13.64780732483729480562617177327, −11.17291795379107930928733238, −10.11899425135237404366041725345, −9.50922390743849164704351272572, −7.8552940614757744428893131345, −7.12004368063868706535417249067, −5.43790404037973794095028211766, −3.889465711054624131419815135916, −2.05025067539745439191742840425,
1.42424761721088411061182516599, 2.44425973129294702577620413051, 4.181102642154717347201579180919, 6.21272097674742442260148895559, 7.86178705713009017605959455933, 8.933910780714109640745696637378, 9.32944252163572748684359047129, 11.38980450761918144478955917664, 12.30260720933460208643142613268, 13.23282751597224192316498910238, 14.46089834892140406284147157942, 16.14905129870592348474881439351, 17.34990837338381108486282229922, 18.261729411894593787316052703154, 19.36728168936878768297765223890, 20.03709292814191507164331983073, 21.298072929050834764110718927936, 21.88320387783206683438138580623, 24.15382862669839461280514097885, 24.62154268147794263929670140281, 25.74307258162342457987635919812, 26.68950245835052688997104881584, 28.04765390382270031208137642705, 28.72390754650391978190683656067, 29.75925567333492757626837437708