L(s) = 1 | + (0.336 + 0.941i)2-s + (0.874 + 0.484i)3-s + (−0.773 + 0.633i)4-s + (−0.619 + 0.785i)5-s + (−0.161 + 0.986i)6-s + (0.837 + 0.546i)7-s + (−0.856 − 0.515i)8-s + (0.530 + 0.847i)9-s + (−0.947 − 0.319i)10-s + (0.796 + 0.605i)11-s + (−0.983 + 0.179i)12-s + (0.958 − 0.284i)13-s + (−0.232 + 0.972i)14-s + (−0.922 + 0.386i)15-s + (0.197 − 0.980i)16-s + (−0.856 + 0.515i)17-s + ⋯ |
L(s) = 1 | + (0.336 + 0.941i)2-s + (0.874 + 0.484i)3-s + (−0.773 + 0.633i)4-s + (−0.619 + 0.785i)5-s + (−0.161 + 0.986i)6-s + (0.837 + 0.546i)7-s + (−0.856 − 0.515i)8-s + (0.530 + 0.847i)9-s + (−0.947 − 0.319i)10-s + (0.796 + 0.605i)11-s + (−0.983 + 0.179i)12-s + (0.958 − 0.284i)13-s + (−0.232 + 0.972i)14-s + (−0.922 + 0.386i)15-s + (0.197 − 0.980i)16-s + (−0.856 + 0.515i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.930 + 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.930 + 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3484299104 + 1.842228352i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3484299104 + 1.842228352i\) |
\(L(1)\) |
\(\approx\) |
\(0.9342802086 + 1.193785505i\) |
\(L(1)\) |
\(\approx\) |
\(0.9342802086 + 1.193785505i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 349 | \( 1 \) |
good | 2 | \( 1 + (0.336 + 0.941i)T \) |
| 3 | \( 1 + (0.874 + 0.484i)T \) |
| 5 | \( 1 + (-0.619 + 0.785i)T \) |
| 7 | \( 1 + (0.837 + 0.546i)T \) |
| 11 | \( 1 + (0.796 + 0.605i)T \) |
| 13 | \( 1 + (0.958 - 0.284i)T \) |
| 17 | \( 1 + (-0.856 + 0.515i)T \) |
| 19 | \( 1 + (-0.0901 - 0.995i)T \) |
| 23 | \( 1 + (0.958 - 0.284i)T \) |
| 29 | \( 1 + (-0.983 - 0.179i)T \) |
| 31 | \( 1 + (-0.994 + 0.108i)T \) |
| 37 | \( 1 + (-0.725 - 0.687i)T \) |
| 41 | \( 1 + (-0.856 - 0.515i)T \) |
| 43 | \( 1 + (0.126 - 0.992i)T \) |
| 47 | \( 1 + (-0.561 - 0.827i)T \) |
| 53 | \( 1 + (0.267 + 0.963i)T \) |
| 59 | \( 1 + (0.874 + 0.484i)T \) |
| 61 | \( 1 + (0.647 + 0.762i)T \) |
| 67 | \( 1 + (0.907 - 0.419i)T \) |
| 71 | \( 1 + (0.874 - 0.484i)T \) |
| 73 | \( 1 + (-0.891 + 0.452i)T \) |
| 79 | \( 1 + (0.267 - 0.963i)T \) |
| 83 | \( 1 + (0.989 + 0.143i)T \) |
| 89 | \( 1 + (-0.302 - 0.953i)T \) |
| 97 | \( 1 + (-0.619 - 0.785i)T \) |
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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
−24.236885912795083399375073943956, −23.70058230090772303992488085512, −22.77789794299300619699933428453, −21.43388050462112942189685169722, −20.592123605738969696042857496421, −20.3014012274714105490279107736, −19.28873244556864954614110260636, −18.63621086724323491445734327475, −17.58320168927961391053899735683, −16.39706333034855064995031558975, −15.07843774740797140369459501764, −14.29403947361317968157503979646, −13.45248303842417113335085573051, −12.79572406231817122966153638253, −11.57405004209402065227699804753, −11.139892810194787963269242140442, −9.533275166381053726037984607061, −8.719492886413033955772503621338, −8.08378856634518482908362949823, −6.68967624040650675304659865206, −5.15986241211698231151793391581, −3.99086813461635893347881524309, −3.4884926625223924911826614476, −1.765305821281118446240280082993, −1.08575152408319322310044569902,
2.16018006092667035057036416666, 3.51383762838593581170555444420, 4.21305388254385324441958035551, 5.30584730528581395778564017421, 6.75761610790688536741880249139, 7.47765161605782350809644581114, 8.66174515280331587435551160599, 8.98767017931686771569075900886, 10.598104727408591858805170692204, 11.53564114661396365261732787892, 12.85922310219809001957942449524, 13.83364952794165801844509598175, 14.826715116589450507048836112238, 15.130256582580225376586947422399, 15.82496506028788049977203726226, 17.123474908199344737050975968788, 18.09823453278649221748638975919, 18.88761234879608307185508214124, 19.979352225404246029423136462172, 20.97250359217621041401430454638, 21.96108215586223909040531630810, 22.46900352572226614815809769723, 23.6043976194625562473159496950, 24.45935147933912690323338345985, 25.297541262439319843993377870872