L(s) = 1 | + (−0.382 − 0.923i)3-s + (0.923 − 0.382i)5-s + (−0.923 − 0.382i)7-s + (−0.707 + 0.707i)9-s + (0.382 − 0.923i)11-s − i·13-s + (−0.707 − 0.707i)15-s + (0.707 + 0.707i)19-s + i·21-s + (−0.382 + 0.923i)23-s + (0.707 − 0.707i)25-s + (0.923 + 0.382i)27-s + (−0.923 + 0.382i)29-s + (0.382 + 0.923i)31-s − 33-s + ⋯ |
L(s) = 1 | + (−0.382 − 0.923i)3-s + (0.923 − 0.382i)5-s + (−0.923 − 0.382i)7-s + (−0.707 + 0.707i)9-s + (0.382 − 0.923i)11-s − i·13-s + (−0.707 − 0.707i)15-s + (0.707 + 0.707i)19-s + i·21-s + (−0.382 + 0.923i)23-s + (0.707 − 0.707i)25-s + (0.923 + 0.382i)27-s + (−0.923 + 0.382i)29-s + (0.382 + 0.923i)31-s − 33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.151 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.151 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6478012267 - 0.5560125721i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6478012267 - 0.5560125721i\) |
\(L(1)\) |
\(\approx\) |
\(0.8620646216 - 0.3982631844i\) |
\(L(1)\) |
\(\approx\) |
\(0.8620646216 - 0.3982631844i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 \) |
good | 3 | \( 1 + (-0.382 - 0.923i)T \) |
| 5 | \( 1 + (0.923 - 0.382i)T \) |
| 7 | \( 1 + (-0.923 - 0.382i)T \) |
| 11 | \( 1 + (0.382 - 0.923i)T \) |
| 13 | \( 1 - iT \) |
| 19 | \( 1 + (0.707 + 0.707i)T \) |
| 23 | \( 1 + (-0.382 + 0.923i)T \) |
| 29 | \( 1 + (-0.923 + 0.382i)T \) |
| 31 | \( 1 + (0.382 + 0.923i)T \) |
| 37 | \( 1 + (0.382 + 0.923i)T \) |
| 41 | \( 1 + (0.923 + 0.382i)T \) |
| 43 | \( 1 + (0.707 - 0.707i)T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 + (-0.707 - 0.707i)T \) |
| 59 | \( 1 + (-0.707 + 0.707i)T \) |
| 61 | \( 1 + (-0.923 - 0.382i)T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + (-0.382 - 0.923i)T \) |
| 73 | \( 1 + (0.923 - 0.382i)T \) |
| 79 | \( 1 + (0.382 - 0.923i)T \) |
| 83 | \( 1 + (-0.707 - 0.707i)T \) |
| 89 | \( 1 + iT \) |
| 97 | \( 1 + (-0.923 + 0.382i)T \) |
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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
−32.47283389138975759028571583215, −31.24843738938456445382085554800, −29.809993979513024094304793121049, −28.646479229293526603568462918530, −28.15695560230549670526673877062, −26.37056905652097320789236698680, −26.00427518854684380169958668214, −24.634813433088680752023151806531, −22.87397581567017718755055909914, −22.23133278041971123855300928151, −21.29605307270859314101915722783, −20.09252286260939911304807332312, −18.57293317917904502605492438819, −17.37028110971568676956208493782, −16.36937675124274696794408378870, −15.17144366965938574862837510956, −14.03661475342933455522818666474, −12.50638684457000217696248769211, −11.134394569372804770052855927198, −9.70069761017559633566695659524, −9.30416549637135546809713705322, −6.80053642666094412020419274071, −5.75470217597515484262362592480, −4.20775425633459956630059228287, −2.50822325962434919927242221689,
1.21256573519733457692217203738, 3.12088639999970769558232313377, 5.51769079212955994020735975917, 6.341732347188707446727123203837, 7.84243656393263407006162122968, 9.38514248613902203299747864325, 10.72436905137983553912868828419, 12.29096584894386513025981571848, 13.271437595523997609778218784021, 14.05557323262944999275292509280, 16.14909218229496937713680239356, 17.08311815344879953899252673597, 18.10299928597700969106392143217, 19.29798119122741077006243333311, 20.36048694361486474817545799846, 21.94790593242770874192318624814, 22.80990217024089429690788556973, 24.11916111431913132643969485681, 25.03597049033551915683813470384, 25.87913331124624835763347511923, 27.47478579506427722589227155939, 28.85357840785085612966895844697, 29.431085604377652844860287224955, 30.21838078632377382643994150301, 31.8264516051641793677904399518