L(s) = 1 | + (−0.546 − 0.837i)3-s + (0.945 − 0.324i)5-s + 7-s + (−0.401 + 0.915i)9-s + (0.245 + 0.969i)11-s + (0.546 − 0.837i)13-s + (−0.789 − 0.614i)15-s + (−0.0825 − 0.996i)17-s + (−0.879 − 0.475i)19-s + (−0.546 − 0.837i)21-s + (0.879 + 0.475i)23-s + (0.789 − 0.614i)25-s + (0.986 − 0.164i)27-s + (−0.789 − 0.614i)29-s + (−0.401 + 0.915i)31-s + ⋯ |
L(s) = 1 | + (−0.546 − 0.837i)3-s + (0.945 − 0.324i)5-s + 7-s + (−0.401 + 0.915i)9-s + (0.245 + 0.969i)11-s + (0.546 − 0.837i)13-s + (−0.789 − 0.614i)15-s + (−0.0825 − 0.996i)17-s + (−0.879 − 0.475i)19-s + (−0.546 − 0.837i)21-s + (0.879 + 0.475i)23-s + (0.789 − 0.614i)25-s + (0.986 − 0.164i)27-s + (−0.789 − 0.614i)29-s + (−0.401 + 0.915i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.492 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 764 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.492 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.427855841 - 0.8326830377i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.427855841 - 0.8326830377i\) |
\(L(1)\) |
\(\approx\) |
\(1.149640675 - 0.3750248596i\) |
\(L(1)\) |
\(\approx\) |
\(1.149640675 - 0.3750248596i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 191 | \( 1 \) |
good | 3 | \( 1 + (-0.546 - 0.837i)T \) |
| 5 | \( 1 + (0.945 - 0.324i)T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + (0.245 + 0.969i)T \) |
| 13 | \( 1 + (0.546 - 0.837i)T \) |
| 17 | \( 1 + (-0.0825 - 0.996i)T \) |
| 19 | \( 1 + (-0.879 - 0.475i)T \) |
| 23 | \( 1 + (0.879 + 0.475i)T \) |
| 29 | \( 1 + (-0.789 - 0.614i)T \) |
| 31 | \( 1 + (-0.401 + 0.915i)T \) |
| 37 | \( 1 + (0.401 + 0.915i)T \) |
| 41 | \( 1 + (0.986 + 0.164i)T \) |
| 43 | \( 1 + (0.677 + 0.735i)T \) |
| 47 | \( 1 + (0.245 + 0.969i)T \) |
| 53 | \( 1 + (-0.245 - 0.969i)T \) |
| 59 | \( 1 + (0.401 - 0.915i)T \) |
| 61 | \( 1 + (0.0825 - 0.996i)T \) |
| 67 | \( 1 + (0.0825 - 0.996i)T \) |
| 71 | \( 1 + (-0.986 - 0.164i)T \) |
| 73 | \( 1 + (-0.245 + 0.969i)T \) |
| 79 | \( 1 + (-0.789 + 0.614i)T \) |
| 83 | \( 1 + (-0.879 + 0.475i)T \) |
| 89 | \( 1 + (0.677 - 0.735i)T \) |
| 97 | \( 1 + (-0.401 - 0.915i)T \) |
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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
−22.25855984578952429387430172534, −21.625351706376173636501933976100, −21.13441766414121816557481955781, −20.55900991036593231232590360397, −19.11612767348076474268134299150, −18.41188346484778173973090412568, −17.49407203664805509618315342661, −16.87557765006522763309451135142, −16.3133358369276190767624992871, −14.89192939894318389630042466634, −14.64551536240420041651853541191, −13.67663260553671592989119941526, −12.65600617663241444434008045024, −11.4349533083183649061186804155, −10.8885011224401121388588601633, −10.34997963468951395342497095103, −8.95709584043434284620353574773, −8.77983189671543181733121906602, −7.19464515654114146260948184193, −5.93296024412356268108135141159, −5.782600101947651762273789950563, −4.43478214045375092029699300535, −3.73455447474185116627678295099, −2.32548187419658127070278408041, −1.22915795304678139086655809048,
1.00202151779673338193198137471, 1.81537090459886091399157885330, 2.72668030397952425220257173435, 4.58778549472482140453557320787, 5.18195066043925321320137435544, 6.076360290956395771511194606416, 7.01686456908635510966683301551, 7.84706912027728820223242689917, 8.807996646104935529021113592000, 9.77060419509821040770428861357, 10.90429485722722725937669257931, 11.41899453541560504098200566420, 12.64294676673286360243620729250, 13.03165573685459099923993113548, 13.98532019588916257891714173808, 14.73982397047410869643422159566, 15.82910656704325686696986563955, 17.01223852826991112513816158665, 17.53889049681616996288304583255, 17.96418539952611468963425285321, 18.79746882178311908686053226677, 19.961224919761035379792640462801, 20.67127705707347805810263795721, 21.39616331233302547857059528911, 22.42366177501763990210502270073