L(s) = 1 | + (−0.374 + 0.927i)2-s + (−0.997 + 0.0697i)3-s + (−0.719 − 0.694i)4-s + (−0.615 + 0.788i)5-s + (0.309 − 0.951i)6-s + (0.961 − 0.275i)7-s + (0.913 − 0.406i)8-s + (0.990 − 0.139i)9-s + (−0.5 − 0.866i)10-s + (0.766 + 0.642i)12-s + (−0.882 + 0.469i)13-s + (−0.104 + 0.994i)14-s + (0.559 − 0.829i)15-s + (0.0348 + 0.999i)16-s + (−0.882 − 0.469i)17-s + (−0.241 + 0.970i)18-s + ⋯ |
L(s) = 1 | + (−0.374 + 0.927i)2-s + (−0.997 + 0.0697i)3-s + (−0.719 − 0.694i)4-s + (−0.615 + 0.788i)5-s + (0.309 − 0.951i)6-s + (0.961 − 0.275i)7-s + (0.913 − 0.406i)8-s + (0.990 − 0.139i)9-s + (−0.5 − 0.866i)10-s + (0.766 + 0.642i)12-s + (−0.882 + 0.469i)13-s + (−0.104 + 0.994i)14-s + (0.559 − 0.829i)15-s + (0.0348 + 0.999i)16-s + (−0.882 − 0.469i)17-s + (−0.241 + 0.970i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.815 - 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.815 - 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3175880542 - 0.1011391980i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3175880542 - 0.1011391980i\) |
\(L(1)\) |
\(\approx\) |
\(0.4673624881 + 0.1781968783i\) |
\(L(1)\) |
\(\approx\) |
\(0.4673624881 + 0.1781968783i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 37 | \( 1 \) |
good | 2 | \( 1 + (-0.374 + 0.927i)T \) |
| 3 | \( 1 + (-0.997 + 0.0697i)T \) |
| 5 | \( 1 + (-0.615 + 0.788i)T \) |
| 7 | \( 1 + (0.961 - 0.275i)T \) |
| 13 | \( 1 + (-0.882 + 0.469i)T \) |
| 17 | \( 1 + (-0.882 - 0.469i)T \) |
| 19 | \( 1 + (-0.997 + 0.0697i)T \) |
| 23 | \( 1 + (-0.5 - 0.866i)T \) |
| 29 | \( 1 + (-0.104 - 0.994i)T \) |
| 31 | \( 1 + (0.309 + 0.951i)T \) |
| 41 | \( 1 + (0.438 - 0.898i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (-0.104 + 0.994i)T \) |
| 53 | \( 1 + (-0.615 - 0.788i)T \) |
| 59 | \( 1 + (0.559 - 0.829i)T \) |
| 61 | \( 1 + (0.990 + 0.139i)T \) |
| 67 | \( 1 + (-0.939 - 0.342i)T \) |
| 71 | \( 1 + (-0.374 - 0.927i)T \) |
| 73 | \( 1 + (-0.809 + 0.587i)T \) |
| 79 | \( 1 + (0.0348 - 0.999i)T \) |
| 83 | \( 1 + (-0.882 - 0.469i)T \) |
| 89 | \( 1 + (-0.939 + 0.342i)T \) |
| 97 | \( 1 + (0.669 - 0.743i)T \) |
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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
−24.10867394995654347353027100589, −23.599851922149248257885053945629, −22.47225054752502443059280332971, −21.72324331270152303019297404911, −21.01381906080151562024166740905, −20.00912987967658064502997003833, −19.29049049548279756200370577988, −18.22746822146663872026153733352, −17.415223007151739455236462120363, −16.97875761116841965145666916985, −15.81655014971213325263295187933, −14.79364127950893110278266714350, −13.22484922743391150629211931943, −12.59057252938526047357009169561, −11.7616447500156709170200545364, −11.1795622473836567375861662647, −10.26415095008456488333782552576, −9.096296214603339727858151613767, −8.14494944296992451007074097509, −7.32800566598940470918622182197, −5.62097658870658687164958955823, −4.680673708826546019768582164343, −4.0696010563124699229201028028, −2.23776007086889480503448696456, −1.16628916289618202070755608776,
0.28906978020757794894371236523, 2.04878039437810700285580763815, 4.28428410483638441525669570122, 4.6467807861878578976767468500, 6.01507051869195097460430854526, 6.886804572470362007950108084460, 7.53588022014528130869463732364, 8.60045614858117721038454905734, 9.95908085429508209578684877623, 10.75787292697028708783741303958, 11.476831092071105783161554918233, 12.58466682144797459300253424382, 14.0037893718278221517739003345, 14.70659627880969195149611632127, 15.57247495411402261639522760061, 16.3497773772780424735314401942, 17.44573137872646979406190563868, 17.754962538506925702519979764204, 18.812308911835928698333229460779, 19.47074926596961330739252664444, 20.92010609767278233773834908293, 22.12462512531093392387363010865, 22.60323726234600251541963490821, 23.596954781257752655912385098989, 24.068467041788847964898649677030