L(s) = 1 | + (0.280 − 0.959i)2-s + (−0.842 − 0.538i)4-s + (−0.599 + 0.800i)5-s + (−0.753 + 0.657i)8-s + (0.599 + 0.800i)10-s + (−0.646 + 0.762i)11-s + (−0.691 − 0.722i)13-s + (0.420 + 0.907i)16-s + (−0.887 − 0.460i)17-s + (−0.978 + 0.207i)19-s + (0.936 − 0.351i)20-s + (0.550 + 0.834i)22-s + (0.163 + 0.986i)23-s + (−0.280 − 0.959i)25-s + (−0.887 + 0.460i)26-s + ⋯ |
L(s) = 1 | + (0.280 − 0.959i)2-s + (−0.842 − 0.538i)4-s + (−0.599 + 0.800i)5-s + (−0.753 + 0.657i)8-s + (0.599 + 0.800i)10-s + (−0.646 + 0.762i)11-s + (−0.691 − 0.722i)13-s + (0.420 + 0.907i)16-s + (−0.887 − 0.460i)17-s + (−0.978 + 0.207i)19-s + (0.936 − 0.351i)20-s + (0.550 + 0.834i)22-s + (0.163 + 0.986i)23-s + (−0.280 − 0.959i)25-s + (−0.887 + 0.460i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.188 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.188 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2692789284 - 0.3258780523i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2692789284 - 0.3258780523i\) |
\(L(1)\) |
\(\approx\) |
\(0.6544876599 - 0.2125583202i\) |
\(L(1)\) |
\(\approx\) |
\(0.6544876599 - 0.2125583202i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 \) |
good | 2 | \( 1 + (0.280 - 0.959i)T \) |
| 5 | \( 1 + (-0.599 + 0.800i)T \) |
| 11 | \( 1 + (-0.646 + 0.762i)T \) |
| 13 | \( 1 + (-0.691 - 0.722i)T \) |
| 17 | \( 1 + (-0.887 - 0.460i)T \) |
| 19 | \( 1 + (-0.978 + 0.207i)T \) |
| 23 | \( 1 + (0.163 + 0.986i)T \) |
| 29 | \( 1 + (-0.963 - 0.266i)T \) |
| 31 | \( 1 + (-0.913 + 0.406i)T \) |
| 37 | \( 1 + (0.251 + 0.967i)T \) |
| 43 | \( 1 + (0.858 - 0.512i)T \) |
| 47 | \( 1 + (0.280 - 0.959i)T \) |
| 53 | \( 1 + (-0.842 - 0.538i)T \) |
| 59 | \( 1 + (0.0149 + 0.999i)T \) |
| 61 | \( 1 + (0.772 + 0.635i)T \) |
| 67 | \( 1 + (0.104 + 0.994i)T \) |
| 71 | \( 1 + (-0.963 + 0.266i)T \) |
| 73 | \( 1 + (-0.826 + 0.563i)T \) |
| 79 | \( 1 + (0.5 + 0.866i)T \) |
| 83 | \( 1 + (-0.900 - 0.433i)T \) |
| 89 | \( 1 + (-0.971 - 0.237i)T \) |
| 97 | \( 1 + (-0.809 + 0.587i)T \) |
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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
−17.61335775639787102850625270550, −17.04908697368334815800979665501, −16.46129348075785221990892013974, −16.04093957366928250617241982149, −15.32518983019944185357812025840, −14.70748132607103385773828776227, −14.11760163746335869452243033211, −13.17658124119901943407023169787, −12.782364846183678071637101896627, −12.332616508106269315397717745231, −11.21117483840563586811811369057, −10.836371292318556378032554420667, −9.53518535000237815326696365023, −9.02401381350577178349267828214, −8.489964733960867224465575384260, −7.78919820580303244241091579830, −7.224968854505346364690147444747, −6.35286223241574657316824074870, −5.74963659909467947460231412089, −4.89602393880207159574744341500, −4.37471498928255010454797560218, −3.823555043488960376269090610825, −2.79917690693418279709457052105, −1.8502022431379159782702210408, −0.45289291839381709088731510188,
0.19357561865155848509766777714, 1.58017363229472455779081917443, 2.447171203879749619101696647452, 2.80849498016249545639492197292, 3.80529758816692430920046855300, 4.2988557391339140610356717815, 5.176706528648222379736197582850, 5.76067526920275217709206402645, 6.876115671748311558777302619152, 7.408753946322084479969209287449, 8.20044580808170595943022475592, 8.99162143532426281021983743837, 9.85332272650567392448865113195, 10.31825260250685762132358469225, 10.96273584070447799100008862670, 11.53918674296759969164240545992, 12.189966673843873887522501022576, 12.962421559481393704102698650655, 13.30688490500313428782059189628, 14.299441287064166011907162442892, 14.95825507008393474775034078157, 15.23538477738513275173901487833, 15.99424544316627436965197233617, 17.213887712845389734458382637723, 17.671148296480557184477061830438