L(s) = 1 | + (−0.5 + 0.866i)7-s + (−0.866 + 0.5i)11-s + (−0.984 − 0.173i)13-s + (−0.939 − 0.342i)17-s + (−0.766 − 0.642i)23-s + (0.342 + 0.939i)29-s + (0.5 − 0.866i)31-s + i·37-s + (−0.173 − 0.984i)41-s + (−0.642 − 0.766i)43-s + (−0.939 + 0.342i)47-s + (−0.5 − 0.866i)49-s + (0.642 − 0.766i)53-s + (0.342 − 0.939i)59-s + (0.642 − 0.766i)61-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)7-s + (−0.866 + 0.5i)11-s + (−0.984 − 0.173i)13-s + (−0.939 − 0.342i)17-s + (−0.766 − 0.642i)23-s + (0.342 + 0.939i)29-s + (0.5 − 0.866i)31-s + i·37-s + (−0.173 − 0.984i)41-s + (−0.642 − 0.766i)43-s + (−0.939 + 0.342i)47-s + (−0.5 − 0.866i)49-s + (0.642 − 0.766i)53-s + (0.342 − 0.939i)59-s + (0.642 − 0.766i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0667i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0667i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8326848308 + 0.02782227713i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8326848308 + 0.02782227713i\) |
\(L(1)\) |
\(\approx\) |
\(0.7764085238 + 0.06623581959i\) |
\(L(1)\) |
\(\approx\) |
\(0.7764085238 + 0.06623581959i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 7 | \( 1 + (-0.5 + 0.866i)T \) |
| 11 | \( 1 + (-0.866 + 0.5i)T \) |
| 13 | \( 1 + (-0.984 - 0.173i)T \) |
| 17 | \( 1 + (-0.939 - 0.342i)T \) |
| 23 | \( 1 + (-0.766 - 0.642i)T \) |
| 29 | \( 1 + (0.342 + 0.939i)T \) |
| 31 | \( 1 + (0.5 - 0.866i)T \) |
| 37 | \( 1 + iT \) |
| 41 | \( 1 + (-0.173 - 0.984i)T \) |
| 43 | \( 1 + (-0.642 - 0.766i)T \) |
| 47 | \( 1 + (-0.939 + 0.342i)T \) |
| 53 | \( 1 + (0.642 - 0.766i)T \) |
| 59 | \( 1 + (0.342 - 0.939i)T \) |
| 61 | \( 1 + (0.642 - 0.766i)T \) |
| 67 | \( 1 + (0.342 + 0.939i)T \) |
| 71 | \( 1 + (-0.766 + 0.642i)T \) |
| 73 | \( 1 + (0.173 + 0.984i)T \) |
| 79 | \( 1 + (-0.173 - 0.984i)T \) |
| 83 | \( 1 + (0.866 + 0.5i)T \) |
| 89 | \( 1 + (-0.173 + 0.984i)T \) |
| 97 | \( 1 + (-0.939 - 0.342i)T \) |
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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
−17.92372158216313636391162173105, −17.78566700151978583330929950075, −16.73705189797480150404315203919, −16.34882878132309168217407728145, −15.58053042896583307145954566828, −14.95396446056747383372046854237, −14.04422147839165063109665473702, −13.49342708452970703118836304965, −12.99839968590570809718047631585, −12.15757677534667573505570153339, −11.4229994809844847169832783013, −10.650443823733066492477159904853, −10.074675876188926066093954468717, −9.51457818863270972062227738643, −8.532511761606339188867808502103, −7.856865215142225125888479768769, −7.20060860321682713753246488759, −6.465674199597017808190129247319, −5.75924290471585754896331653341, −4.78872075458456861036035567676, −4.23502717692025101463539075587, −3.306847411565384929900719294307, −2.604219035476115182947206609259, −1.70796641731431437506834387847, −0.51789686686601776016954962733,
0.393656201555766215480030305674, 1.98573054275813401262096942788, 2.42683480768324582864922915212, 3.156364968780250788746198735836, 4.23225131521449969981167654522, 5.0468424441069735742036223029, 5.49086815578762285892296961980, 6.59800432501754979524160851991, 6.96062416848686787769118866429, 8.04961524020096542860459507277, 8.50817975346482607383242172218, 9.45731505806798020342750104335, 9.970729563520172999912464683458, 10.60192504576043150291254028173, 11.640866577989831373125087835503, 12.10495979172985703561792895362, 12.88113932031966577730190341870, 13.28551937443078675640047081846, 14.312398755466831273507124052770, 14.931093438131657254630448221809, 15.63511578916255557091536485120, 15.997824728006949732414996825, 16.90484192399982486146276229217, 17.66679259900437720465319520973, 18.20479282279067149535653565603