L(s) = 1 | + (0.273 − 0.961i)2-s + (−0.183 + 0.982i)3-s + (−0.850 − 0.526i)4-s + (0.961 − 0.273i)5-s + (0.895 + 0.445i)6-s + (−0.0922 − 0.995i)7-s + (−0.739 + 0.673i)8-s + (−0.932 − 0.361i)9-s − i·10-s + (0.850 + 0.526i)11-s + (0.673 − 0.739i)12-s + (0.995 − 0.0922i)13-s + (−0.982 − 0.183i)14-s + (0.0922 + 0.995i)15-s + (0.445 + 0.895i)16-s + (−0.739 − 0.673i)17-s + ⋯ |
L(s) = 1 | + (0.273 − 0.961i)2-s + (−0.183 + 0.982i)3-s + (−0.850 − 0.526i)4-s + (0.961 − 0.273i)5-s + (0.895 + 0.445i)6-s + (−0.0922 − 0.995i)7-s + (−0.739 + 0.673i)8-s + (−0.932 − 0.361i)9-s − i·10-s + (0.850 + 0.526i)11-s + (0.673 − 0.739i)12-s + (0.995 − 0.0922i)13-s + (−0.982 − 0.183i)14-s + (0.0922 + 0.995i)15-s + (0.445 + 0.895i)16-s + (−0.739 − 0.673i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.449 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.449 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.050680812 - 0.6477462349i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.050680812 - 0.6477462349i\) |
\(L(1)\) |
\(\approx\) |
\(1.103739747 - 0.4387916953i\) |
\(L(1)\) |
\(\approx\) |
\(1.103739747 - 0.4387916953i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 137 | \( 1 \) |
good | 2 | \( 1 + (0.273 - 0.961i)T \) |
| 3 | \( 1 + (-0.183 + 0.982i)T \) |
| 5 | \( 1 + (0.961 - 0.273i)T \) |
| 7 | \( 1 + (-0.0922 - 0.995i)T \) |
| 11 | \( 1 + (0.850 + 0.526i)T \) |
| 13 | \( 1 + (0.995 - 0.0922i)T \) |
| 17 | \( 1 + (-0.739 - 0.673i)T \) |
| 19 | \( 1 + (0.602 - 0.798i)T \) |
| 23 | \( 1 + (0.895 - 0.445i)T \) |
| 29 | \( 1 + (-0.895 + 0.445i)T \) |
| 31 | \( 1 + (-0.798 + 0.602i)T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + iT \) |
| 43 | \( 1 + (0.798 + 0.602i)T \) |
| 47 | \( 1 + (-0.361 + 0.932i)T \) |
| 53 | \( 1 + (-0.798 - 0.602i)T \) |
| 59 | \( 1 + (0.932 + 0.361i)T \) |
| 61 | \( 1 + (-0.932 + 0.361i)T \) |
| 67 | \( 1 + (-0.995 + 0.0922i)T \) |
| 71 | \( 1 + (0.526 + 0.850i)T \) |
| 73 | \( 1 + (0.0922 - 0.995i)T \) |
| 79 | \( 1 + (0.183 + 0.982i)T \) |
| 83 | \( 1 + (-0.673 - 0.739i)T \) |
| 89 | \( 1 + (-0.961 + 0.273i)T \) |
| 97 | \( 1 + (-0.526 + 0.850i)T \) |
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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
−28.73114483311955976274052363320, −27.693541472070080299514721757487, −26.20366271932351818323619872531, −25.3980469644521838339834239332, −24.76458167425045780460382503033, −24.014976640959937834205912452168, −22.643182430431965120661602393975, −22.12941978839640144827844107260, −20.95058929265420837156180508806, −19.04168990642909985868290190222, −18.42744088086596202054006658575, −17.48791880951145502966801712854, −16.66073368389530520612183334817, −15.2631012880261769895732218002, −14.1611986012249640681912392123, −13.40483252356175891068315126039, −12.46420321944944421871919418223, −11.19268329556744385529000700350, −9.23534876103424628335094918308, −8.53529008074839481621394094868, −7.04773481527938297878386188269, −6.02501802716225839707570284402, −5.59237572218287518052635844796, −3.4316143080861803070154705994, −1.750052715616100354849405751686,
1.274506003190145530913782576210, 3.066788928966200977120360290518, 4.27973322416900175920550033274, 5.186097861251758635862057969191, 6.58455977696924836743295900420, 8.96880424512111358813894607839, 9.49880853800904237328538199711, 10.65091320894134863286255941539, 11.3247871676648911022415992123, 12.89400648381929627333931506194, 13.81812043523539439274670860624, 14.68038966359152195725681997536, 16.19549037882883455905592308082, 17.32396646713927300114871923664, 18.02865014502738554475827865748, 19.79331334198679906299054531402, 20.48437231718874535938998953655, 21.13810669959412463362978730732, 22.34624547948749300594368975046, 22.77233119802382082388764303045, 24.11356053329890651655341101396, 25.61513052452389059794677728290, 26.58407519091643531505333801791, 27.525916970840375621860967759700, 28.44263018908058431170243717632