L(s) = 1 | + (0.654 + 0.755i)5-s + (−0.841 − 0.540i)7-s + (−0.142 − 0.989i)11-s + (0.841 − 0.540i)13-s + (0.959 − 0.281i)17-s + (0.959 + 0.281i)19-s + (−0.142 + 0.989i)25-s + (0.959 − 0.281i)29-s + (−0.415 + 0.909i)31-s + (−0.142 − 0.989i)35-s + (−0.654 + 0.755i)37-s + (0.654 + 0.755i)41-s + (−0.415 − 0.909i)43-s + 47-s + (0.415 + 0.909i)49-s + ⋯ |
L(s) = 1 | + (0.654 + 0.755i)5-s + (−0.841 − 0.540i)7-s + (−0.142 − 0.989i)11-s + (0.841 − 0.540i)13-s + (0.959 − 0.281i)17-s + (0.959 + 0.281i)19-s + (−0.142 + 0.989i)25-s + (0.959 − 0.281i)29-s + (−0.415 + 0.909i)31-s + (−0.142 − 0.989i)35-s + (−0.654 + 0.755i)37-s + (0.654 + 0.755i)41-s + (−0.415 − 0.909i)43-s + 47-s + (0.415 + 0.909i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.328737077 - 0.08717245512i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.328737077 - 0.08717245512i\) |
\(L(1)\) |
\(\approx\) |
\(1.152365792 + 0.01560872508i\) |
\(L(1)\) |
\(\approx\) |
\(1.152365792 + 0.01560872508i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 23 | \( 1 \) |
good | 5 | \( 1 + (0.654 + 0.755i)T \) |
| 7 | \( 1 + (-0.841 - 0.540i)T \) |
| 11 | \( 1 + (-0.142 - 0.989i)T \) |
| 13 | \( 1 + (0.841 - 0.540i)T \) |
| 17 | \( 1 + (0.959 - 0.281i)T \) |
| 19 | \( 1 + (0.959 + 0.281i)T \) |
| 29 | \( 1 + (0.959 - 0.281i)T \) |
| 31 | \( 1 + (-0.415 + 0.909i)T \) |
| 37 | \( 1 + (-0.654 + 0.755i)T \) |
| 41 | \( 1 + (0.654 + 0.755i)T \) |
| 43 | \( 1 + (-0.415 - 0.909i)T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 + (-0.841 - 0.540i)T \) |
| 59 | \( 1 + (0.841 - 0.540i)T \) |
| 61 | \( 1 + (0.415 - 0.909i)T \) |
| 67 | \( 1 + (0.142 - 0.989i)T \) |
| 71 | \( 1 + (-0.142 + 0.989i)T \) |
| 73 | \( 1 + (-0.959 - 0.281i)T \) |
| 79 | \( 1 + (-0.841 + 0.540i)T \) |
| 83 | \( 1 + (-0.654 + 0.755i)T \) |
| 89 | \( 1 + (-0.415 - 0.909i)T \) |
| 97 | \( 1 + (-0.654 - 0.755i)T \) |
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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
−25.60002124381452224636208487078, −24.99504499395009936183014864630, −23.88623117053579634674233444751, −22.97733634033433019766230527652, −22.022522591837023829577107002228, −21.08677120105013818544640703160, −20.36820773322217968956098074178, −19.30369383455795420882442721560, −18.305598196079883553430635720077, −17.45200732363273905295706600892, −16.33398517498314943129994883859, −15.77539126812542272066548553983, −14.46592026102207193023099114761, −13.43002644136421695422951355726, −12.61177733066319422430322919968, −11.84599847850775344095912340800, −10.294960532841422506148380395519, −9.48142893926699530161859370947, −8.73708879451138363003331927850, −7.3576985081853750118274619511, −6.13145037292049337749470511009, −5.32287243993783016458586259726, −4.03025622486615248821521169384, −2.6183003441845657733636311583, −1.32738014525038000705084377054,
1.13311554044004588705649740250, 2.989320564097240540355245808052, 3.5094230113028027019234361292, 5.411914693966189442083085605, 6.23004391863059678232520364337, 7.21505721771558235399136476072, 8.42782345389639144352583502834, 9.74923478759935330812352663394, 10.39243788207569687888803359773, 11.360648362188466890035986722419, 12.69911651745718631580747389440, 13.75986606073473831336046137360, 14.14921087916073851411758397483, 15.64678500487244620097156361214, 16.35476653330774523980272324609, 17.42196053498092314663314395016, 18.457229561789986901703493772, 19.05913885894626873556529640477, 20.239076335465846455793563260584, 21.17314522465902481311482007871, 22.10339090714172572194303470757, 22.892277178672799521198894889647, 23.65288002325750973719743454028, 25.04833235447191832938476716945, 25.58724559095099592066663794412