L(s) = 1 | + (0.0307 − 0.999i)2-s + (−0.998 − 0.0615i)4-s + (0.779 − 0.626i)5-s + (0.816 − 0.577i)7-s + (−0.0922 + 0.995i)8-s + (−0.602 − 0.798i)10-s + (0.0307 − 0.999i)11-s + (−0.908 − 0.417i)13-s + (−0.552 − 0.833i)14-s + (0.992 + 0.122i)16-s + (−0.739 − 0.673i)17-s + (−0.273 − 0.961i)19-s + (−0.816 + 0.577i)20-s + (−0.998 − 0.0615i)22-s + (0.0307 + 0.999i)23-s + ⋯ |
L(s) = 1 | + (0.0307 − 0.999i)2-s + (−0.998 − 0.0615i)4-s + (0.779 − 0.626i)5-s + (0.816 − 0.577i)7-s + (−0.0922 + 0.995i)8-s + (−0.602 − 0.798i)10-s + (0.0307 − 0.999i)11-s + (−0.908 − 0.417i)13-s + (−0.552 − 0.833i)14-s + (0.992 + 0.122i)16-s + (−0.739 − 0.673i)17-s + (−0.273 − 0.961i)19-s + (−0.816 + 0.577i)20-s + (−0.998 − 0.0615i)22-s + (0.0307 + 0.999i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.237 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.237 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.7345464047 - 0.9357114930i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.7345464047 - 0.9357114930i\) |
\(L(1)\) |
\(\approx\) |
\(0.6537234906 - 0.7873283303i\) |
\(L(1)\) |
\(\approx\) |
\(0.6537234906 - 0.7873283303i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 103 | \( 1 \) |
good | 2 | \( 1 + (0.0307 - 0.999i)T \) |
| 5 | \( 1 + (0.779 - 0.626i)T \) |
| 7 | \( 1 + (0.816 - 0.577i)T \) |
| 11 | \( 1 + (0.0307 - 0.999i)T \) |
| 13 | \( 1 + (-0.908 - 0.417i)T \) |
| 17 | \( 1 + (-0.739 - 0.673i)T \) |
| 19 | \( 1 + (-0.273 - 0.961i)T \) |
| 23 | \( 1 + (0.0307 + 0.999i)T \) |
| 29 | \( 1 + (0.153 + 0.988i)T \) |
| 31 | \( 1 + (-0.389 - 0.920i)T \) |
| 37 | \( 1 + (-0.982 + 0.183i)T \) |
| 41 | \( 1 + (0.153 - 0.988i)T \) |
| 43 | \( 1 + (0.332 + 0.943i)T \) |
| 47 | \( 1 + (0.5 + 0.866i)T \) |
| 53 | \( 1 + (0.273 + 0.961i)T \) |
| 59 | \( 1 + (-0.816 - 0.577i)T \) |
| 61 | \( 1 + (0.213 - 0.976i)T \) |
| 67 | \( 1 + (0.816 + 0.577i)T \) |
| 71 | \( 1 + (-0.932 + 0.361i)T \) |
| 73 | \( 1 + (0.932 - 0.361i)T \) |
| 79 | \( 1 + (-0.153 - 0.988i)T \) |
| 83 | \( 1 + (-0.816 + 0.577i)T \) |
| 89 | \( 1 + (-0.445 + 0.895i)T \) |
| 97 | \( 1 + (-0.952 - 0.303i)T \) |
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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
−22.38520256703250816795690377746, −21.54657454832314675321001348606, −21.01227305682851730639041387118, −19.70256325683274869692738988491, −18.68108199356965006515986455764, −18.15128598182866064126600366925, −17.34836581373028569756191152986, −16.96957015127385366800788803769, −15.693131272626146182800670180034, −14.80061707661306231252661544193, −14.61704051738562963886902165413, −13.72674055467984119222781407798, −12.668724317276815747570152763838, −12.008075535421352789958352933680, −10.60706219437208440615669993268, −9.958190940333824913170794316486, −9.03418924896920926925968839434, −8.2523792036545217601336799601, −7.23029675312469939238290071715, −6.58056742985702872708728462571, −5.66501404844506245257326844781, −4.855265090955058529568910249692, −4.01411420657009059738345943340, −2.444935733573003375041027412321, −1.68006684888690317309572889766,
0.25359441298303150473596000205, 1.127191138236804281886320891467, 2.119802523717100568097221339976, 3.03793645697062560865278585036, 4.29704311744183114316790923488, 5.01846051340027088530273666783, 5.70012332370130396317755842567, 7.165359404023102256174231821117, 8.22890899307469575139954939154, 9.04543895053716072343407454301, 9.668279678495006222560322800448, 10.78023231878624489249461847950, 11.19608188698686002050927488497, 12.24256951277177093423080755564, 13.088111383899570654518565972574, 13.79242141770257726985325471596, 14.25024387662787322906111899105, 15.45255520142658345471598442058, 16.67992152782526004219520434980, 17.47741001261323256384598624074, 17.75537592212323842808299055548, 18.85788808229123626503971173728, 19.787874935678191926620776326664, 20.3317743115825403503311195677, 21.05380430894837218881091760482