L(s) = 1 | + (−0.993 + 0.111i)2-s + (0.875 + 0.483i)3-s + (0.974 − 0.222i)4-s + (−0.0560 + 0.998i)5-s + (−0.923 − 0.382i)6-s + (−0.382 + 0.923i)7-s + (−0.943 + 0.330i)8-s + (0.532 + 0.846i)9-s + (−0.0560 − 0.998i)10-s + (−0.578 + 0.815i)11-s + (0.960 + 0.276i)12-s + (−0.433 + 0.900i)13-s + (0.276 − 0.960i)14-s + (−0.532 + 0.846i)15-s + (0.900 − 0.433i)16-s + ⋯ |
L(s) = 1 | + (−0.993 + 0.111i)2-s + (0.875 + 0.483i)3-s + (0.974 − 0.222i)4-s + (−0.0560 + 0.998i)5-s + (−0.923 − 0.382i)6-s + (−0.382 + 0.923i)7-s + (−0.943 + 0.330i)8-s + (0.532 + 0.846i)9-s + (−0.0560 − 0.998i)10-s + (−0.578 + 0.815i)11-s + (0.960 + 0.276i)12-s + (−0.433 + 0.900i)13-s + (0.276 − 0.960i)14-s + (−0.532 + 0.846i)15-s + (0.900 − 0.433i)16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0514i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0514i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.02399194084 + 0.9312905357i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.02399194084 + 0.9312905357i\) |
\(L(1)\) |
\(\approx\) |
\(0.6263880865 + 0.5297898860i\) |
\(L(1)\) |
\(\approx\) |
\(0.6263880865 + 0.5297898860i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 \) |
| 43 | \( 1 \) |
good | 2 | \( 1 + (-0.993 + 0.111i)T \) |
| 3 | \( 1 + (0.875 + 0.483i)T \) |
| 5 | \( 1 + (-0.0560 + 0.998i)T \) |
| 7 | \( 1 + (-0.382 + 0.923i)T \) |
| 11 | \( 1 + (-0.578 + 0.815i)T \) |
| 13 | \( 1 + (-0.433 + 0.900i)T \) |
| 19 | \( 1 + (-0.532 + 0.846i)T \) |
| 23 | \( 1 + (0.578 - 0.815i)T \) |
| 29 | \( 1 + (0.960 + 0.276i)T \) |
| 31 | \( 1 + (0.276 - 0.960i)T \) |
| 37 | \( 1 + (0.923 - 0.382i)T \) |
| 41 | \( 1 + (-0.483 - 0.875i)T \) |
| 47 | \( 1 + (-0.974 + 0.222i)T \) |
| 53 | \( 1 + (-0.943 - 0.330i)T \) |
| 59 | \( 1 + (0.330 - 0.943i)T \) |
| 61 | \( 1 + (0.960 - 0.276i)T \) |
| 67 | \( 1 + (0.222 + 0.974i)T \) |
| 71 | \( 1 + (-0.578 - 0.815i)T \) |
| 73 | \( 1 + (-0.0560 + 0.998i)T \) |
| 79 | \( 1 + (0.923 + 0.382i)T \) |
| 83 | \( 1 + (0.111 - 0.993i)T \) |
| 89 | \( 1 + (0.781 - 0.623i)T \) |
| 97 | \( 1 + (0.985 - 0.167i)T \) |
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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
−21.65969231725314095488471673873, −20.98970175420193653722975022393, −20.227611394622918801140043353403, −19.58000134290866518037972036594, −19.2638958791987952274241304146, −17.97413650063031123817233628908, −17.41443171329026580669985688698, −16.49045289135662089568463007069, −15.75527901839050151316447903870, −14.95966592351055920792765115610, −13.57333731191000363649953216496, −13.10308965531965250194169749066, −12.282110268985786843694949533782, −11.15820878928833064358765590721, −10.141815352128278161250033907052, −9.45366249442877026591563004394, −8.47203909153497556972925230448, −8.01094349546005336024715855740, −7.15012142643556707891129300871, −6.191089257192639032256609938817, −4.79185033001784460881875072924, −3.41061830179124654686430224368, −2.721012130540083855165424551010, −1.32555329771827985374125579791, −0.548753419052283207259562431279,
2.10059766785163214762843549534, 2.40763776781300314105554908872, 3.45938736303245881728220114279, 4.81232325628935933641571056163, 6.18791734867258639870226627076, 6.97617611839716982737780571238, 7.87700500439057238057562355813, 8.65986788336019695308576635407, 9.64871790033787151317725950608, 10.05833939107209812306098947770, 10.975408244743951418837095525133, 11.98594155947291094101049776505, 12.95750908871926419203946925823, 14.53576833333423571177285203878, 14.684263045548879253565826898536, 15.63392389608529892520278128107, 16.200425995930574819418981039113, 17.31419793628301958783048149056, 18.37577302795725538791911596008, 18.90950432686075242079610704375, 19.359771511883466692596247803525, 20.42646378842615765451899525235, 21.17053020157896983177957133697, 21.86520312995048484741731729731, 22.82241707780314584776975168426