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
−29.75925567333492757626837437708, −28.72390754650391978190683656067, −28.04765390382270031208137642705, −26.68950245835052688997104881584, −25.74307258162342457987635919812, −24.62154268147794263929670140281, −24.15382862669839461280514097885, −21.88320387783206683438138580623, −21.298072929050834764110718927936, −20.03709292814191507164331983073, −19.36728168936878768297765223890, −18.261729411894593787316052703154, −17.34990837338381108486282229922, −16.14905129870592348474881439351, −14.46089834892140406284147157942, −13.23282751597224192316498910238, −12.30260720933460208643142613268, −11.38980450761918144478955917664, −9.32944252163572748684359047129, −8.933910780714109640745696637378, −7.86178705713009017605959455933, −6.21272097674742442260148895559, −4.181102642154717347201579180919, −2.44425973129294702577620413051, −1.42424761721088411061182516599,
2.05025067539745439191742840425, 3.889465711054624131419815135916, 5.43790404037973794095028211766, 7.12004368063868706535417249067, 7.8552940614757744428893131345, 9.50922390743849164704351272572, 10.11899425135237404366041725345, 11.17291795379107930928733238, 13.64780732483729480562617177327, 14.478412085917524888864397045924, 15.15964467399975329404166496834, 16.47517316450189496952207731392, 17.52333832230022617729391634820, 18.509666923888102186386022538425, 19.93429397633467418410409920976, 20.434811608950944189587028804936, 22.202987429283951341343987228655, 22.94143832395573040500156432147, 24.542793782147726802592123386604, 25.43871712012455923740443959595, 26.22082981262359043762837255012, 27.17736489934507641243826333153, 27.637818687680838942685756234820, 29.36036750644000330700385790599, 30.28912443694643183419912464213