L(s) = 1 | + (−0.691 + 0.722i)2-s + (0.473 − 0.880i)3-s + (−0.0448 − 0.998i)4-s + (0.858 − 0.512i)5-s + (0.309 + 0.951i)6-s + (−0.809 − 0.587i)7-s + (0.753 + 0.657i)8-s + (−0.550 − 0.834i)9-s + (−0.222 + 0.974i)10-s + (−0.900 − 0.433i)12-s + (−0.995 − 0.0896i)13-s + (0.983 − 0.178i)14-s + (−0.0448 − 0.998i)15-s + (−0.995 + 0.0896i)16-s + (−0.995 + 0.0896i)17-s + (0.983 + 0.178i)18-s + ⋯ |
L(s) = 1 | + (−0.691 + 0.722i)2-s + (0.473 − 0.880i)3-s + (−0.0448 − 0.998i)4-s + (0.858 − 0.512i)5-s + (0.309 + 0.951i)6-s + (−0.809 − 0.587i)7-s + (0.753 + 0.657i)8-s + (−0.550 − 0.834i)9-s + (−0.222 + 0.974i)10-s + (−0.900 − 0.433i)12-s + (−0.995 − 0.0896i)13-s + (0.983 − 0.178i)14-s + (−0.0448 − 0.998i)15-s + (−0.995 + 0.0896i)16-s + (−0.995 + 0.0896i)17-s + (0.983 + 0.178i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 473 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.789 - 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 473 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.789 - 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2150842293 - 0.6268942709i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2150842293 - 0.6268942709i\) |
\(L(1)\) |
\(\approx\) |
\(0.6983581045 - 0.2517012359i\) |
\(L(1)\) |
\(\approx\) |
\(0.6983581045 - 0.2517012359i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 43 | \( 1 \) |
good | 2 | \( 1 + (-0.691 + 0.722i)T \) |
| 3 | \( 1 + (0.473 - 0.880i)T \) |
| 5 | \( 1 + (0.858 - 0.512i)T \) |
| 7 | \( 1 + (-0.809 - 0.587i)T \) |
| 13 | \( 1 + (-0.995 - 0.0896i)T \) |
| 17 | \( 1 + (-0.995 + 0.0896i)T \) |
| 19 | \( 1 + (-0.963 - 0.266i)T \) |
| 23 | \( 1 + (0.623 - 0.781i)T \) |
| 29 | \( 1 + (0.473 + 0.880i)T \) |
| 31 | \( 1 + (-0.691 + 0.722i)T \) |
| 37 | \( 1 + (-0.809 - 0.587i)T \) |
| 41 | \( 1 + (0.983 - 0.178i)T \) |
| 47 | \( 1 + (-0.963 - 0.266i)T \) |
| 53 | \( 1 + (0.858 + 0.512i)T \) |
| 59 | \( 1 + (0.753 - 0.657i)T \) |
| 61 | \( 1 + (-0.691 - 0.722i)T \) |
| 67 | \( 1 + (0.623 + 0.781i)T \) |
| 71 | \( 1 + (-0.550 + 0.834i)T \) |
| 73 | \( 1 + (-0.393 + 0.919i)T \) |
| 79 | \( 1 + (0.309 - 0.951i)T \) |
| 83 | \( 1 + (-0.691 - 0.722i)T \) |
| 89 | \( 1 + (-0.900 + 0.433i)T \) |
| 97 | \( 1 + (-0.550 - 0.834i)T \) |
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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
−24.62296734681580214971297692517, −22.59749951359258467008075472127, −22.33422338619800312881670677007, −21.35501261055456227094037291484, −21.03951975226278107776252216405, −19.61654291584626037576453274199, −19.394806979837821095372077601571, −18.292283165348137570119577519406, −17.29790502770727377519685289064, −16.67291823073728382749579955605, −15.554643163426728137508694529726, −14.7983951239145810880370305167, −13.5700022393191657496546681943, −12.90955931977849560050521333295, −11.6574878324879472547120496628, −10.707506212369115493273314873057, −9.907360299721151099188622039361, −9.36913576680210397678185203376, −8.62480099554815713319818973556, −7.322689112499674422939113275584, −6.19331970698924011265517576047, −4.865456371699765041742793069236, −3.66404144130216573360757325846, −2.62655155881255170386386295576, −2.123049882631405528393289329052,
0.39654015689063501061262678832, 1.71412514083725501520617416798, 2.69060263146496523215074320301, 4.507435245857870271601633201008, 5.699157017663253629725227060285, 6.77373366090862138429721760465, 7.06468322899244890370240257362, 8.509442710097529670971430402476, 9.02316690775854004990819019097, 9.9633081092960492036111725613, 10.873831930897191074071128065814, 12.588952459586458563609556742155, 13.05880290199631895953521535873, 14.07942036004643801701776374818, 14.693255263297030600760092094876, 15.915927979994836330504839404175, 16.89854617456866280610828582666, 17.45059247069610838774833379115, 18.19318822163208469584160808671, 19.31683167202329633991397909057, 19.76768544570878845772961946785, 20.56247688281543729244000818039, 21.8993055180054817128936603037, 23.00082297548304831327947955085, 23.78280874063695826654335242020