L(s) = 1 | + (0.990 + 0.136i)2-s + (0.113 + 0.993i)3-s + (0.962 + 0.269i)4-s + (0.158 + 0.987i)5-s + (−0.0227 + 0.999i)6-s + (0.934 − 0.356i)7-s + (0.917 + 0.398i)8-s + (−0.974 + 0.225i)9-s + (0.0227 + 0.999i)10-s + (−0.898 − 0.439i)11-s + (−0.158 + 0.987i)12-s + (0.682 − 0.730i)13-s + (0.974 − 0.225i)14-s + (−0.962 + 0.269i)15-s + (0.854 + 0.519i)16-s + (0.648 + 0.761i)17-s + ⋯ |
L(s) = 1 | + (0.990 + 0.136i)2-s + (0.113 + 0.993i)3-s + (0.962 + 0.269i)4-s + (0.158 + 0.987i)5-s + (−0.0227 + 0.999i)6-s + (0.934 − 0.356i)7-s + (0.917 + 0.398i)8-s + (−0.974 + 0.225i)9-s + (0.0227 + 0.999i)10-s + (−0.898 − 0.439i)11-s + (−0.158 + 0.987i)12-s + (0.682 − 0.730i)13-s + (0.974 − 0.225i)14-s + (−0.962 + 0.269i)15-s + (0.854 + 0.519i)16-s + (0.648 + 0.761i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.107 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.107 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.867084365 + 1.675577905i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.867084365 + 1.675577905i\) |
\(L(1)\) |
\(\approx\) |
\(1.782450326 + 0.9505791774i\) |
\(L(1)\) |
\(\approx\) |
\(1.782450326 + 0.9505791774i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 277 | \( 1 \) |
good | 2 | \( 1 + (0.990 + 0.136i)T \) |
| 3 | \( 1 + (0.113 + 0.993i)T \) |
| 5 | \( 1 + (0.158 + 0.987i)T \) |
| 7 | \( 1 + (0.934 - 0.356i)T \) |
| 11 | \( 1 + (-0.898 - 0.439i)T \) |
| 13 | \( 1 + (0.682 - 0.730i)T \) |
| 17 | \( 1 + (0.648 + 0.761i)T \) |
| 19 | \( 1 + (-0.775 - 0.631i)T \) |
| 23 | \( 1 + (-0.158 - 0.987i)T \) |
| 29 | \( 1 + (-0.715 - 0.699i)T \) |
| 31 | \( 1 + (-0.934 - 0.356i)T \) |
| 37 | \( 1 + (-0.962 - 0.269i)T \) |
| 41 | \( 1 + (-0.334 + 0.942i)T \) |
| 43 | \( 1 + (-0.613 + 0.789i)T \) |
| 47 | \( 1 + (0.983 - 0.181i)T \) |
| 53 | \( 1 + (0.949 + 0.313i)T \) |
| 59 | \( 1 + (-0.0682 + 0.997i)T \) |
| 61 | \( 1 + (0.0682 - 0.997i)T \) |
| 67 | \( 1 + (-0.877 + 0.480i)T \) |
| 71 | \( 1 + (0.803 - 0.595i)T \) |
| 73 | \( 1 + (-0.854 + 0.519i)T \) |
| 79 | \( 1 + (0.934 + 0.356i)T \) |
| 83 | \( 1 + (0.0227 - 0.999i)T \) |
| 89 | \( 1 + (0.995 + 0.0909i)T \) |
| 97 | \( 1 + (-0.113 - 0.993i)T \) |
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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
−25.38056143145622359309071411487, −24.18414408115677474422487394351, −23.78369041141283356611916194778, −23.11754979615713153992768640044, −21.66579986055319100017650319978, −20.74083402168348982429370818532, −20.43832716392751649623803737458, −19.09777522745824800472820912920, −18.25278563433526209954349092490, −17.1125011400798715974148206251, −16.13223042896383238554432396208, −15.00404226620096437666610425904, −13.98172980108710804723291298109, −13.341838192537744304337994217953, −12.329436656359124163591229335828, −11.83671351524996906867550579864, −10.68051789169629118614514072394, −9.03020062673446005295745754354, −7.97930234360341466049368422391, −7.0897756265675959860583913874, −5.62526855928553573820276031705, −5.16910768583301541457389676505, −3.72869155807745828990815387654, −2.110049167445458178132968528934, −1.50050231132447972925351351921,
2.22107239906454405466132116911, 3.27494119459527625569128527906, 4.17595836182905517439950081573, 5.35171672464056516104630901991, 6.133931226050900958262469753996, 7.61206238132059138954053575231, 8.43398916518628442637076037188, 10.41667257012853706318305845518, 10.68087687264058156773903751769, 11.52841635472142049872644498685, 13.08915985619761269521010148806, 13.96660174141035440056511215399, 14.91380684138062445935967540734, 15.23958264745481295281938319482, 16.43565329011658126240796620260, 17.35587559286237000514792970010, 18.54507751200092547303331385111, 19.87813329183492643196111692380, 20.86858968058106144294895965523, 21.33723755805952209541970887475, 22.174191449699605581569029934, 23.1094506150783061277789555119, 23.725787883756887298769103772636, 24.98465566007502200414636829447, 26.08696840816827523435064446503