L(s) = 1 | + (−0.438 + 0.898i)2-s + (−0.615 − 0.788i)4-s + (−0.997 − 0.0697i)5-s + (−0.0348 + 0.999i)7-s + (0.978 − 0.207i)8-s + (0.5 − 0.866i)10-s + (0.719 − 0.694i)13-s + (−0.882 − 0.469i)14-s + (−0.241 + 0.970i)16-s + (0.104 + 0.994i)17-s + (0.978 − 0.207i)19-s + (0.559 + 0.829i)20-s + (−0.939 + 0.342i)23-s + (0.990 + 0.139i)25-s + (0.309 + 0.951i)26-s + ⋯ |
L(s) = 1 | + (−0.438 + 0.898i)2-s + (−0.615 − 0.788i)4-s + (−0.997 − 0.0697i)5-s + (−0.0348 + 0.999i)7-s + (0.978 − 0.207i)8-s + (0.5 − 0.866i)10-s + (0.719 − 0.694i)13-s + (−0.882 − 0.469i)14-s + (−0.241 + 0.970i)16-s + (0.104 + 0.994i)17-s + (0.978 − 0.207i)19-s + (0.559 + 0.829i)20-s + (−0.939 + 0.342i)23-s + (0.990 + 0.139i)25-s + (0.309 + 0.951i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.954 - 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.954 - 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.08303450572 + 0.5424119900i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.08303450572 + 0.5424119900i\) |
\(L(1)\) |
\(\approx\) |
\(0.5492094991 + 0.3435326916i\) |
\(L(1)\) |
\(\approx\) |
\(0.5492094991 + 0.3435326916i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.438 + 0.898i)T \) |
| 5 | \( 1 + (-0.997 - 0.0697i)T \) |
| 7 | \( 1 + (-0.0348 + 0.999i)T \) |
| 13 | \( 1 + (0.719 - 0.694i)T \) |
| 17 | \( 1 + (0.104 + 0.994i)T \) |
| 19 | \( 1 + (0.978 - 0.207i)T \) |
| 23 | \( 1 + (-0.939 + 0.342i)T \) |
| 29 | \( 1 + (0.882 - 0.469i)T \) |
| 31 | \( 1 + (0.961 + 0.275i)T \) |
| 37 | \( 1 + (-0.978 - 0.207i)T \) |
| 41 | \( 1 + (0.882 + 0.469i)T \) |
| 43 | \( 1 + (-0.766 + 0.642i)T \) |
| 47 | \( 1 + (-0.615 + 0.788i)T \) |
| 53 | \( 1 + (-0.809 - 0.587i)T \) |
| 59 | \( 1 + (-0.374 + 0.927i)T \) |
| 61 | \( 1 + (-0.961 + 0.275i)T \) |
| 67 | \( 1 + (0.173 - 0.984i)T \) |
| 71 | \( 1 + (-0.104 - 0.994i)T \) |
| 73 | \( 1 + (-0.669 + 0.743i)T \) |
| 79 | \( 1 + (-0.438 + 0.898i)T \) |
| 83 | \( 1 + (0.719 + 0.694i)T \) |
| 89 | \( 1 + (-0.5 - 0.866i)T \) |
| 97 | \( 1 + (-0.997 + 0.0697i)T \) |
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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
−24.659275752462002523111375345987, −23.44747412138005192357491522621, −22.932190277237827308742405367613, −21.96435271266489960296693425448, −20.625781379667671727483516528204, −20.30950928744284854663115369685, −19.299708042281354946574770561197, −18.55030035714189308394401647912, −17.60972874522278103879908921939, −16.40728326142911661723139159182, −15.92066392226434155363661751320, −14.15450184575219626861926741161, −13.61479482582885350901908234270, −12.230680790667590609941863648422, −11.60434622603311448524298360029, −10.699167875219386390026656605781, −9.784602805291083269487779885520, −8.59465718484631382445779107272, −7.70254227810162844622805966792, −6.808985100722892177043424946767, −4.78801690435353309838359941899, −3.911938307288204558013187120, −3.03455543972536006937457067408, −1.33643346362206567348846294828, −0.23069069093370158199245018501,
1.27732119793271094968967701733, 3.18679331608317562659105307197, 4.49787509823480756866063503368, 5.63585320458504077347935497772, 6.517219677563252819518766019095, 7.9464087201774456843325405034, 8.29249101528229932082253144687, 9.436812766562474958665120684527, 10.58072964453453103077659526142, 11.72007534408190405053624517303, 12.71923955582142735029175304976, 13.92931694563148651070258163809, 15.05652007798157069179599955359, 15.6720206806713768358795294627, 16.23204942451255452456020242490, 17.61262066433825056056210472509, 18.26722345523943841171519878876, 19.2703525664099664303920681799, 19.856067090565021141353726940565, 21.24438327353404778135491204532, 22.47042207422870438960859822743, 23.06822165745826861966123995629, 24.10462347142592583359862992448, 24.69908058667044387428250482057, 25.708235150771184501605554020815