L(s) = 1 | + (0.749 + 0.662i)2-s + (0.123 + 0.992i)4-s + (−0.991 − 0.132i)5-s + (0.345 − 0.938i)7-s + (−0.564 + 0.825i)8-s + (−0.654 − 0.755i)10-s + (−0.683 − 0.730i)13-s + (0.879 − 0.475i)14-s + (−0.969 + 0.244i)16-s + (−0.362 + 0.931i)17-s + (0.774 − 0.633i)19-s + (0.00951 − 0.999i)20-s + (−0.786 + 0.618i)23-s + (0.964 + 0.263i)25-s + (−0.0285 − 0.999i)26-s + ⋯ |
L(s) = 1 | + (0.749 + 0.662i)2-s + (0.123 + 0.992i)4-s + (−0.991 − 0.132i)5-s + (0.345 − 0.938i)7-s + (−0.564 + 0.825i)8-s + (−0.654 − 0.755i)10-s + (−0.683 − 0.730i)13-s + (0.879 − 0.475i)14-s + (−0.969 + 0.244i)16-s + (−0.362 + 0.931i)17-s + (0.774 − 0.633i)19-s + (0.00951 − 0.999i)20-s + (−0.786 + 0.618i)23-s + (0.964 + 0.263i)25-s + (−0.0285 − 0.999i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.838 - 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.838 - 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.329750130 - 0.3937856180i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.329750130 - 0.3937856180i\) |
\(L(1)\) |
\(\approx\) |
\(1.187244834 + 0.2102084600i\) |
\(L(1)\) |
\(\approx\) |
\(1.187244834 + 0.2102084600i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.749 + 0.662i)T \) |
| 5 | \( 1 + (-0.991 - 0.132i)T \) |
| 7 | \( 1 + (0.345 - 0.938i)T \) |
| 13 | \( 1 + (-0.683 - 0.730i)T \) |
| 17 | \( 1 + (-0.362 + 0.931i)T \) |
| 19 | \( 1 + (0.774 - 0.633i)T \) |
| 23 | \( 1 + (-0.786 + 0.618i)T \) |
| 29 | \( 1 + (-0.710 - 0.703i)T \) |
| 31 | \( 1 + (0.820 - 0.572i)T \) |
| 37 | \( 1 + (0.516 - 0.856i)T \) |
| 41 | \( 1 + (-0.595 - 0.803i)T \) |
| 43 | \( 1 + (0.723 - 0.690i)T \) |
| 47 | \( 1 + (0.953 - 0.299i)T \) |
| 53 | \( 1 + (0.696 - 0.717i)T \) |
| 59 | \( 1 + (-0.398 - 0.917i)T \) |
| 61 | \( 1 + (-0.948 - 0.318i)T \) |
| 67 | \( 1 + (0.580 - 0.814i)T \) |
| 71 | \( 1 + (-0.998 + 0.0570i)T \) |
| 73 | \( 1 + (0.897 - 0.441i)T \) |
| 79 | \( 1 + (-0.761 + 0.647i)T \) |
| 83 | \( 1 + (0.999 + 0.0380i)T \) |
| 89 | \( 1 + (0.841 - 0.540i)T \) |
| 97 | \( 1 + (-0.991 + 0.132i)T \) |
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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
−21.68424851738267081958798679860, −20.62093018835667980335779484379, −20.13237498493719794770579133127, −19.251524151608305077251260400153, −18.57128525705170004046204727277, −18.07967498552597583560254318116, −16.51399545715254874972219627311, −15.87703261849942970067387722525, −15.07436698015340655435430341654, −14.463777445068023044654958352455, −13.72609889469734378621667550661, −12.52674361648159661618130937349, −11.912920294019199122136425686755, −11.59782731740337691978565660127, −10.60710841348669983053284415577, −9.60031142880351925738471822210, −8.82474322249726173633546837237, −7.72780799422002176360206025381, −6.815592403170474149331444246999, −5.820940552829641229658624136007, −4.821889220261113243970996706979, −4.291800938284839694801958481178, −3.09350393493744159781115889296, −2.456687824664447303420619853582, −1.24276700642988567268580482945,
0.46774127266053155232977780039, 2.212482169252861104684302692, 3.48592532656262791505329175195, 4.042961566457654022769629254266, 4.8585511386446856456865919533, 5.76196107605425076573640759452, 6.901378038927756881428832276630, 7.68056156559522480250151146304, 7.98118973653013345341424071157, 9.14111999390062069243614207750, 10.39850983360048519710358514005, 11.27995456851803623908674048155, 11.97994340971412043864484650314, 12.80855221796195529624332955434, 13.58026709794781708423721351855, 14.33595596593483255398330746018, 15.350544304094249704753049142218, 15.54818383966180763341297240410, 16.691743244256685316601108718262, 17.24768416396617508864588346048, 17.95376368655747836031238843503, 19.242130346485222085427713232359, 20.07109507540604208097236297855, 20.464244574891661212043692149170, 21.53229042596338559455252772173