L(s) = 1 | + (−0.861 + 0.508i)2-s + (0.483 − 0.875i)4-s + (−0.179 − 0.983i)5-s + (0.830 − 0.556i)7-s + (0.0285 + 0.999i)8-s + (0.654 + 0.755i)10-s + (0.905 − 0.424i)13-s + (−0.432 + 0.901i)14-s + (−0.532 − 0.846i)16-s + (0.998 + 0.0570i)17-s + (0.254 − 0.967i)19-s + (−0.948 − 0.318i)20-s + (−0.786 + 0.618i)23-s + (−0.935 + 0.353i)25-s + (−0.564 + 0.825i)26-s + ⋯ |
L(s) = 1 | + (−0.861 + 0.508i)2-s + (0.483 − 0.875i)4-s + (−0.179 − 0.983i)5-s + (0.830 − 0.556i)7-s + (0.0285 + 0.999i)8-s + (0.654 + 0.755i)10-s + (0.905 − 0.424i)13-s + (−0.432 + 0.901i)14-s + (−0.532 − 0.846i)16-s + (0.998 + 0.0570i)17-s + (0.254 − 0.967i)19-s + (−0.948 − 0.318i)20-s + (−0.786 + 0.618i)23-s + (−0.935 + 0.353i)25-s + (−0.564 + 0.825i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.933 - 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.933 - 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1538773313 - 0.8299137417i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1538773313 - 0.8299137417i\) |
\(L(1)\) |
\(\approx\) |
\(0.7247316762 - 0.1660896865i\) |
\(L(1)\) |
\(\approx\) |
\(0.7247316762 - 0.1660896865i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.861 + 0.508i)T \) |
| 5 | \( 1 + (-0.179 - 0.983i)T \) |
| 7 | \( 1 + (0.830 - 0.556i)T \) |
| 13 | \( 1 + (0.905 - 0.424i)T \) |
| 17 | \( 1 + (0.998 + 0.0570i)T \) |
| 19 | \( 1 + (0.254 - 0.967i)T \) |
| 23 | \( 1 + (-0.786 + 0.618i)T \) |
| 29 | \( 1 + (-0.161 - 0.986i)T \) |
| 31 | \( 1 + (-0.290 - 0.956i)T \) |
| 37 | \( 1 + (-0.921 + 0.389i)T \) |
| 41 | \( 1 + (-0.953 - 0.299i)T \) |
| 43 | \( 1 + (-0.723 + 0.690i)T \) |
| 47 | \( 1 + (-0.595 + 0.803i)T \) |
| 53 | \( 1 + (-0.466 - 0.884i)T \) |
| 59 | \( 1 + (-0.217 + 0.976i)T \) |
| 61 | \( 1 + (-0.00951 + 0.999i)T \) |
| 67 | \( 1 + (0.580 - 0.814i)T \) |
| 71 | \( 1 + (-0.362 - 0.931i)T \) |
| 73 | \( 1 + (0.985 + 0.170i)T \) |
| 79 | \( 1 + (-0.380 - 0.924i)T \) |
| 83 | \( 1 + (-0.272 - 0.962i)T \) |
| 89 | \( 1 + (0.841 - 0.540i)T \) |
| 97 | \( 1 + (-0.179 + 0.983i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−21.52227986572292091643962058946, −20.731948522936097731796028737520, −20.04945932367737202547062193317, −18.92093519775923822029407618819, −18.47268260905731358446361065109, −18.12200241134541626882818103267, −17.07286138390508516172758969995, −16.20321846447943953492774079012, −15.50672142265243327772149958062, −14.45812495853065430934506608437, −13.949706024799128274436367857529, −12.49707090694606270633912055515, −11.92653084449511303735439158594, −11.16139149692400550902022536814, −10.49382152718420325182580182401, −9.768875057874908500115618213307, −8.58718270022393180325215890340, −8.14252896191383939689046105599, −7.17942538471645553700951211733, −6.36867187901553492225393791483, −5.28083297260312076265239505031, −3.80348739098783047399618906852, −3.2491030489855964719595863089, −2.04269646303985142838054626060, −1.382111614747665682862151692976,
0.245746660051939964152425621580, 1.132437072097854789191727488965, 1.86535437103633731825431119038, 3.491203123887680421253817305926, 4.64973298646012608985951583389, 5.39956337232325289667259294446, 6.23877795242144061533182767274, 7.51995674170802334505084181353, 7.97289461256466043264399704218, 8.68682273170615729236763090313, 9.59748892300591795061950484699, 10.36079081722288999646707117018, 11.38230226364846548525996815497, 11.85536392570978277854815026350, 13.256926929597694043302194507573, 13.81655538996565741813102901905, 14.88923828751792834195775242305, 15.610903947661555761295678745504, 16.335474035236309427308772503913, 17.08450860801661576220663790983, 17.64111862093173681004431889765, 18.41139077497208451273234983852, 19.364835793511698421565754947411, 20.10496333023633361256800404419, 20.70829965881608610045892848251