L(s) = 1 | + (0.809 + 0.587i)2-s − 3-s + (0.309 + 0.951i)4-s + (−0.809 − 0.587i)6-s + (−0.809 + 0.587i)7-s + (−0.309 + 0.951i)8-s + 9-s + (0.809 + 0.587i)11-s + (−0.309 − 0.951i)12-s + (0.309 + 0.951i)13-s − 14-s + (−0.809 + 0.587i)16-s + (0.309 + 0.951i)17-s + (0.809 + 0.587i)18-s + (−0.309 + 0.951i)19-s + ⋯ |
L(s) = 1 | + (0.809 + 0.587i)2-s − 3-s + (0.309 + 0.951i)4-s + (−0.809 − 0.587i)6-s + (−0.809 + 0.587i)7-s + (−0.309 + 0.951i)8-s + 9-s + (0.809 + 0.587i)11-s + (−0.309 − 0.951i)12-s + (0.309 + 0.951i)13-s − 14-s + (−0.809 + 0.587i)16-s + (0.309 + 0.951i)17-s + (0.809 + 0.587i)18-s + (−0.309 + 0.951i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0286 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0286 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.4649673912 + 0.4518360447i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.4649673912 + 0.4518360447i\) |
\(L(1)\) |
\(\approx\) |
\(0.7365026024 + 0.6615715721i\) |
\(L(1)\) |
\(\approx\) |
\(0.7365026024 + 0.6615715721i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 241 | \( 1 \) |
good | 2 | \( 1 + (0.809 + 0.587i)T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + (-0.809 + 0.587i)T \) |
| 11 | \( 1 + (0.809 + 0.587i)T \) |
| 13 | \( 1 + (0.309 + 0.951i)T \) |
| 17 | \( 1 + (0.309 + 0.951i)T \) |
| 19 | \( 1 + (-0.309 + 0.951i)T \) |
| 23 | \( 1 + (-0.809 + 0.587i)T \) |
| 29 | \( 1 + (0.309 - 0.951i)T \) |
| 31 | \( 1 + (-0.309 - 0.951i)T \) |
| 37 | \( 1 + (-0.809 - 0.587i)T \) |
| 41 | \( 1 + (0.309 - 0.951i)T \) |
| 43 | \( 1 + (-0.809 - 0.587i)T \) |
| 47 | \( 1 + (-0.309 + 0.951i)T \) |
| 53 | \( 1 + (-0.309 + 0.951i)T \) |
| 59 | \( 1 + (-0.809 - 0.587i)T \) |
| 61 | \( 1 + (-0.809 + 0.587i)T \) |
| 67 | \( 1 + (-0.309 - 0.951i)T \) |
| 71 | \( 1 + (-0.309 + 0.951i)T \) |
| 73 | \( 1 + (0.309 - 0.951i)T \) |
| 79 | \( 1 + (-0.809 + 0.587i)T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 + (0.809 + 0.587i)T \) |
| 97 | \( 1 + (-0.309 + 0.951i)T \) |
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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
−17.09235334513211445792669251983, −16.41109068948071521363795049299, −15.970275782930452346361777302455, −15.355358261358145630489857007389, −14.377526996078586638823611877144, −13.83748690554160620294004575102, −12.98327515411111704072864830333, −12.78704206608762388597956444260, −11.77786637373246211527153248176, −11.517115614556195286875633175490, −10.48712007944454292639427360778, −10.3692109526902876776266443440, −9.51355619736599001186185428727, −8.72294019111820808715732611330, −7.4860191115155990541149940894, −6.61717941704468649889530112564, −6.47508100747417244672791289131, −5.56499185772670673059737625277, −4.90636588077965981394191933073, −4.27277348991195104088150592577, −3.31310524786653788725583170091, −3.02799537839338220648803456980, −1.63741437063971532469108918541, −0.9244609800445526718997036062, −0.155050427121417140033809411164,
1.6254801434061547023359853964, 2.096831048683361986148712461717, 3.4650030364342588552867344670, 4.00641552651075997648449393038, 4.50281951839739341858995909226, 5.62135363019482141979703502863, 6.04137533695971052612190463028, 6.42833816905460005386790840452, 7.212509699554324957121531840943, 7.91065610793505173917434499982, 8.89099012687162373882100475433, 9.56565686589898199158093275508, 10.30486464552802649155863803364, 11.21291151037736351230897307517, 11.92421602008391279743167952678, 12.29291343413963449246874546888, 12.75465257796194317642031050121, 13.65630961214126569233432254958, 14.23432542591241483097448243919, 15.16807882233105291463226559920, 15.52945206479300019563996373203, 16.28060716804366090028975943026, 16.81732075811479340243795245080, 17.24565451918189487613245965268, 17.978635956538972408366257634619