L(s) = 1 | + (0.5 − 0.866i)5-s + (0.5 + 0.866i)7-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)13-s − 17-s − 19-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + (0.5 + 0.866i)29-s + (0.5 − 0.866i)31-s + 35-s + 37-s + (0.5 − 0.866i)41-s + (0.5 + 0.866i)43-s + (−0.5 − 0.866i)47-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)5-s + (0.5 + 0.866i)7-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)13-s − 17-s − 19-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + (0.5 + 0.866i)29-s + (0.5 − 0.866i)31-s + 35-s + 37-s + (0.5 − 0.866i)41-s + (0.5 + 0.866i)43-s + (−0.5 − 0.866i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8222820649 - 0.07194035890i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8222820649 - 0.07194035890i\) |
\(L(1)\) |
\(\approx\) |
\(1.003837166 - 0.05783774636i\) |
\(L(1)\) |
\(\approx\) |
\(1.003837166 - 0.05783774636i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (0.5 + 0.866i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 + (-0.5 + 0.866i)T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 + (-0.5 + 0.866i)T \) |
| 29 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 + (0.5 - 0.866i)T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + (0.5 - 0.866i)T \) |
| 43 | \( 1 + (0.5 + 0.866i)T \) |
| 47 | \( 1 + (-0.5 - 0.866i)T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 + (-0.5 + 0.866i)T \) |
| 61 | \( 1 + (-0.5 - 0.866i)T \) |
| 67 | \( 1 + (0.5 - 0.866i)T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 + (0.5 + 0.866i)T \) |
| 83 | \( 1 + (-0.5 - 0.866i)T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + (-0.5 - 0.866i)T \) |
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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
−36.00525677501136349756372020648, −34.32559475708389148454345625691, −33.59985372037602719440421343101, −32.410377116432291659288084035111, −30.709460580858837860147969780235, −30.00072573157095372216919796400, −28.70783783763675238999960959202, −27.137992170147889144606728655683, −26.19713634504464078781560156483, −24.94746180270863493885699970595, −23.41316604449680327709302166251, −22.36886154269478372052879267445, −20.96480270317158436691406526239, −19.73252620095592696928190534912, −18.050841047001815437271326540424, −17.29453928386547325791063122901, −15.34094671625210298882315145436, −14.22256159954633033580274083309, −12.85216184693912363313832213407, −10.91146184578109010597210076996, −10.01002603795048827922562440232, −7.881531411731270516392470116776, −6.5261700824006092798773189112, −4.55075499877039430801681565928, −2.41511530935627412534947040506,
2.14409489942574708228920090773, 4.676075508748655458490821851313, 6.06203944896421696376204364589, 8.278342933690077090023168122797, 9.35390753944101506765847727587, 11.25279823282029673209891658310, 12.648325869272542811061859051250, 13.97502385043088870723119736977, 15.58071229483743724297034263903, 16.889791040762615134076772847028, 18.169039935900764697177121392176, 19.586675629076211234362596893133, 21.18031806028247547499431076424, 21.79998852975647480543378572569, 23.870700345723942497068926763896, 24.59162420280926346667433058825, 25.93193174639440572944941739710, 27.43679296486427561565187471753, 28.57261072631808318262395139322, 29.5081392852163109266857295296, 31.27588267842812972985599510952, 31.980390955335435978492680125656, 33.43701017854223800184568857670, 34.48935673580116156526442706543, 35.86690603875864830574326672474