L(s) = 1 | + (0.997 + 0.0647i)2-s + (0.271 − 0.962i)3-s + (0.991 + 0.129i)4-s + (0.665 − 0.746i)5-s + (0.333 − 0.942i)6-s + (−0.735 + 0.677i)7-s + (0.981 + 0.193i)8-s + (−0.852 − 0.523i)9-s + (0.712 − 0.701i)10-s + (0.145 − 0.989i)11-s + (0.393 − 0.919i)12-s + (−0.481 + 0.876i)13-s + (−0.777 + 0.628i)14-s + (−0.536 − 0.843i)15-s + (0.966 + 0.256i)16-s + (0.712 − 0.701i)17-s + ⋯ |
L(s) = 1 | + (0.997 + 0.0647i)2-s + (0.271 − 0.962i)3-s + (0.991 + 0.129i)4-s + (0.665 − 0.746i)5-s + (0.333 − 0.942i)6-s + (−0.735 + 0.677i)7-s + (0.981 + 0.193i)8-s + (−0.852 − 0.523i)9-s + (0.712 − 0.701i)10-s + (0.145 − 0.989i)11-s + (0.393 − 0.919i)12-s + (−0.481 + 0.876i)13-s + (−0.777 + 0.628i)14-s + (−0.536 − 0.843i)15-s + (0.966 + 0.256i)16-s + (0.712 − 0.701i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 389 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.353 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 389 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.353 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.256197109 - 1.559123714i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.256197109 - 1.559123714i\) |
\(L(1)\) |
\(\approx\) |
\(1.950722982 - 0.7567197255i\) |
\(L(1)\) |
\(\approx\) |
\(1.950722982 - 0.7567197255i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 389 | \( 1 \) |
good | 2 | \( 1 + (0.997 + 0.0647i)T \) |
| 3 | \( 1 + (0.271 - 0.962i)T \) |
| 5 | \( 1 + (0.665 - 0.746i)T \) |
| 7 | \( 1 + (-0.735 + 0.677i)T \) |
| 11 | \( 1 + (0.145 - 0.989i)T \) |
| 13 | \( 1 + (-0.481 + 0.876i)T \) |
| 17 | \( 1 + (0.712 - 0.701i)T \) |
| 19 | \( 1 + (-0.302 - 0.953i)T \) |
| 23 | \( 1 + (0.665 + 0.746i)T \) |
| 29 | \( 1 + (-0.995 + 0.0970i)T \) |
| 31 | \( 1 + (0.925 + 0.378i)T \) |
| 37 | \( 1 + (0.452 - 0.891i)T \) |
| 41 | \( 1 + (-0.302 + 0.953i)T \) |
| 43 | \( 1 + (-0.884 + 0.466i)T \) |
| 47 | \( 1 + (-0.302 + 0.953i)T \) |
| 53 | \( 1 + (0.208 + 0.977i)T \) |
| 59 | \( 1 + (-0.937 + 0.348i)T \) |
| 61 | \( 1 + (-0.481 + 0.876i)T \) |
| 67 | \( 1 + (-0.986 - 0.161i)T \) |
| 71 | \( 1 + (0.991 + 0.129i)T \) |
| 73 | \( 1 + (0.966 - 0.256i)T \) |
| 79 | \( 1 + (0.925 - 0.378i)T \) |
| 83 | \( 1 + (-0.986 + 0.161i)T \) |
| 89 | \( 1 + (-0.957 - 0.287i)T \) |
| 97 | \( 1 + (0.991 - 0.129i)T \) |
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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
−24.84571589502388626443062723635, −23.29220632501693603184369536934, −22.64223362874755646282530139763, −22.28846074876745162826449733634, −21.18442004509081285309042212013, −20.54929078565986780709039265882, −19.77343051320985589940650340558, −18.77081672208075016715610017762, −17.0269108039457314484889336785, −16.83156312640038892151197051565, −15.30072684755124363756940540145, −14.95548981405743386761080997241, −14.105673121761318723830806031268, −13.18384397707934419020350596174, −12.28968226633196153835408841737, −10.9112091102503746562348392035, −10.10993801251736086750412917122, −9.85346892091333153886833436640, −7.974821485312959817147504529569, −6.88638372694038937556640767130, −5.93870194006095621407435286210, −4.95095336108434688787372091949, −3.79820442488297574560250079241, −3.110730424318626235196761798411, −2.002232329910730187661498340824,
1.25379561465331380440704950852, 2.46338101653674062978820560806, 3.23490953175931253245820037904, 4.847490631912245758521007590803, 5.82691353658386783835397469510, 6.47243639129479518595875889819, 7.526227050816954272681340391328, 8.806575713444038615814633552955, 9.5439814568972514057719663635, 11.2920425526014106067880304397, 12.03207479833172614414925531002, 12.8664876437521087393084445566, 13.52007609982976477665415402100, 14.17378855930537381967602585286, 15.27689903319174773741387449848, 16.45739927082330526862876324266, 16.9636135328383867056960390982, 18.30917986425365907385659355311, 19.337607593827459545624691191915, 19.83353414018555446034808875052, 21.1571258924292067189697502713, 21.56540399550226698216300323312, 22.638260767063028184418820047042, 23.617522869324505249841929618290, 24.34165526464002573642779296493