L(s) = 1 | + 2-s − i·3-s + 4-s − i·6-s − 7-s + 8-s − 9-s + i·11-s − i·12-s − 14-s + 16-s − i·17-s − 18-s + i·19-s + i·21-s + i·22-s + ⋯ |
L(s) = 1 | + 2-s − i·3-s + 4-s − i·6-s − 7-s + 8-s − 9-s + i·11-s − i·12-s − 14-s + 16-s − i·17-s − 18-s + i·19-s + i·21-s + i·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.370223614 - 0.5182767396i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.370223614 - 0.5182767396i\) |
\(L(1)\) |
\(\approx\) |
\(1.509020967 - 0.3934177522i\) |
\(L(1)\) |
\(\approx\) |
\(1.509020967 - 0.3934177522i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 - iT \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 \) |
| 37 | \( 1 + iT \) |
| 41 | \( 1 - iT \) |
| 43 | \( 1 \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 - iT \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + iT \) |
| 73 | \( 1 \) |
| 79 | \( 1 + iT \) |
| 83 | \( 1 + iT \) |
| 89 | \( 1 + iT \) |
| 97 | \( 1 - iT \) |
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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
−32.41853970777710965853987771892, −31.5758704623335932514376905170, −30.25796209793602351175812045235, −29.025443496265801308883278279476, −28.201631990661722202553890557307, −26.52083949374780001601147821427, −25.78732605882320498162277696973, −24.384732968951227329665134656671, −23.13844817907205670553600961034, −22.11685481116609652099043754270, −21.48043337481769483807996723489, −20.188764571321683522745992388904, −19.223035996993874458235487904242, −16.937287913756485956335891842757, −16.0912078599684542591068481651, −15.13876593942303081398713292265, −13.91165143025147292765528815983, −12.71520260351638053829560741744, −11.23379464835647882019909554642, −10.24367035547808864852972342042, −8.657566982297700079000366442275, −6.59275479542356088164165191934, −5.443838858444506126727971709034, −3.9481074453100913983968138706, −2.90412928662050702441472068580,
2.021772809675842337552614596094, 3.50131320946130805990772191375, 5.43063343920782791257483816656, 6.681703304601103134456438392618, 7.62449161914241303510973803601, 9.7362230751714873125686076329, 11.51603978912734431588818044882, 12.539180530275797883243500302988, 13.34959280200627499809914976107, 14.52411608373154521630155619692, 15.87555225541885527284388745111, 17.15407976702283749818794367538, 18.70245836868697524493959515176, 19.794610609711137157839077243726, 20.7573108308320364486410852110, 22.5710746238596995168529542523, 22.92684144059871440778320203568, 24.20244069896380307661487297243, 25.260895045787242570960807399733, 25.92799470888533723557828048136, 28.100492331750352999294686111205, 29.3055513243783886595917395257, 29.77160546293504586129371889553, 31.18225832852193301584617457271, 31.67672906426039715577274133851