L(s) = 1 | + (−0.674 − 0.738i)2-s + (0.997 − 0.0721i)3-s + (−0.0901 + 0.995i)4-s + (0.935 − 0.353i)5-s + (−0.725 − 0.687i)6-s + (0.874 + 0.484i)7-s + (0.796 − 0.605i)8-s + (0.989 − 0.143i)9-s + (−0.891 − 0.452i)10-s + (0.647 − 0.762i)11-s + (−0.0180 + 0.999i)12-s + (0.267 − 0.963i)13-s + (−0.232 − 0.972i)14-s + (0.907 − 0.419i)15-s + (−0.983 − 0.179i)16-s + (−0.994 + 0.108i)17-s + ⋯ |
L(s) = 1 | + (−0.674 − 0.738i)2-s + (0.997 − 0.0721i)3-s + (−0.0901 + 0.995i)4-s + (0.935 − 0.353i)5-s + (−0.725 − 0.687i)6-s + (0.874 + 0.484i)7-s + (0.796 − 0.605i)8-s + (0.989 − 0.143i)9-s + (−0.891 − 0.452i)10-s + (0.647 − 0.762i)11-s + (−0.0180 + 0.999i)12-s + (0.267 − 0.963i)13-s + (−0.232 − 0.972i)14-s + (0.907 − 0.419i)15-s + (−0.983 − 0.179i)16-s + (−0.994 + 0.108i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0856 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0856 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.033389240 - 1.866062459i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.033389240 - 1.866062459i\) |
\(L(1)\) |
\(\approx\) |
\(1.355218831 - 0.6413888077i\) |
\(L(1)\) |
\(\approx\) |
\(1.355218831 - 0.6413888077i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 4003 | \( 1 \) |
good | 2 | \( 1 + (-0.674 - 0.738i)T \) |
| 3 | \( 1 + (0.997 - 0.0721i)T \) |
| 5 | \( 1 + (0.935 - 0.353i)T \) |
| 7 | \( 1 + (0.874 + 0.484i)T \) |
| 11 | \( 1 + (0.647 - 0.762i)T \) |
| 13 | \( 1 + (0.267 - 0.963i)T \) |
| 17 | \( 1 + (-0.994 + 0.108i)T \) |
| 19 | \( 1 + (0.590 + 0.806i)T \) |
| 23 | \( 1 + (-0.983 - 0.179i)T \) |
| 29 | \( 1 + (0.590 - 0.806i)T \) |
| 31 | \( 1 + (-0.817 - 0.576i)T \) |
| 37 | \( 1 + (0.590 - 0.806i)T \) |
| 41 | \( 1 + (0.647 + 0.762i)T \) |
| 43 | \( 1 + (-0.436 - 0.899i)T \) |
| 47 | \( 1 + (-0.983 + 0.179i)T \) |
| 53 | \( 1 + (-0.983 + 0.179i)T \) |
| 59 | \( 1 + (0.126 + 0.992i)T \) |
| 61 | \( 1 + (0.958 + 0.284i)T \) |
| 67 | \( 1 + (-0.232 + 0.972i)T \) |
| 71 | \( 1 + (0.997 - 0.0721i)T \) |
| 73 | \( 1 + (-0.947 + 0.319i)T \) |
| 79 | \( 1 + (0.958 - 0.284i)T \) |
| 83 | \( 1 + (-0.674 - 0.738i)T \) |
| 89 | \( 1 + (-0.232 - 0.972i)T \) |
| 97 | \( 1 + (-0.232 + 0.972i)T \) |
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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
−18.31793681991868802945664503686, −17.96708859823673130020265545242, −17.50272504646386255208983093143, −16.6016942530978949166812556108, −15.93512787553116329299407895236, −15.13637295131627473609965517879, −14.48666727816421793496138149065, −14.06815452239288527933814691647, −13.66592114636763647177774430175, −12.76563399546180442241146582096, −11.43633391254441715426888929908, −10.91502906719432340875184940700, −9.989102870451659456315839387786, −9.50912791630846301375369317375, −8.95583504571477651543171718383, −8.267264780374077426906983752, −7.409643056759772648371608516303, −6.80166934714109530401713472644, −6.39884654444641454908992351060, −4.99805489484944306626065787962, −4.65988944522254842534414972716, −3.67125236664636761164815829493, −2.38190938026769529153018859900, −1.76794146447993580090109419639, −1.277121540888474027638620289334,
0.86248611144077579611054518011, 1.6222442809328931814260338751, 2.22369333965967946989735855291, 2.90917534142523806093329023017, 3.8516357630014275299224700707, 4.50436732663324945270804235947, 5.63787954200667032189997289276, 6.35775324690833158046704293766, 7.52772443526267400552130219351, 8.19914120972151582068315815343, 8.599578591271704989748929975731, 9.27061715764492042329214746794, 9.88452943015120689945338735053, 10.582915890259190267294399833129, 11.39408246913929005477298611947, 12.12218532629007024544304326481, 12.95358117379908102576993809640, 13.39997712896662378317172932246, 14.17458869267668736217544254115, 14.678443210815455251329492602380, 15.77115881314335703184805894895, 16.301291011623930043087174545032, 17.26134284576070559777951082944, 17.91040451714937825736277306788, 18.25123801514437667752231957295