L(s) = 1 | − 5-s + (0.5 + 0.866i)7-s + (−0.5 − 0.866i)11-s + (−0.5 − 0.866i)13-s + (0.5 + 0.866i)17-s + (−0.5 − 0.866i)23-s + 25-s − 29-s + (0.5 − 0.866i)31-s + (−0.5 − 0.866i)35-s + 37-s − 41-s + (0.5 − 0.866i)43-s + 47-s + (−0.5 + 0.866i)49-s + ⋯ |
L(s) = 1 | − 5-s + (0.5 + 0.866i)7-s + (−0.5 − 0.866i)11-s + (−0.5 − 0.866i)13-s + (0.5 + 0.866i)17-s + (−0.5 − 0.866i)23-s + 25-s − 29-s + (0.5 − 0.866i)31-s + (−0.5 − 0.866i)35-s + 37-s − 41-s + (0.5 − 0.866i)43-s + 47-s + (−0.5 + 0.866i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.532 - 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.532 - 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8409184102 - 0.4642541097i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8409184102 - 0.4642541097i\) |
\(L(1)\) |
\(\approx\) |
\(0.8711243809 - 0.08570864036i\) |
\(L(1)\) |
\(\approx\) |
\(0.8711243809 - 0.08570864036i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 \) |
good | 5 | \( 1 - T \) |
| 7 | \( 1 + (0.5 + 0.866i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 + (-0.5 - 0.866i)T \) |
| 17 | \( 1 + (0.5 + 0.866i)T \) |
| 23 | \( 1 + (-0.5 - 0.866i)T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 + (0.5 - 0.866i)T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 + (0.5 - 0.866i)T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 + (0.5 - 0.866i)T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 + (0.5 + 0.866i)T \) |
| 71 | \( 1 + (-0.5 - 0.866i)T \) |
| 73 | \( 1 + (-0.5 - 0.866i)T \) |
| 79 | \( 1 + (0.5 - 0.866i)T \) |
| 83 | \( 1 + (-0.5 - 0.866i)T \) |
| 89 | \( 1 + (0.5 - 0.866i)T \) |
| 97 | \( 1 + (-0.5 + 0.866i)T \) |
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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
−23.12679884764348470285024093569, −22.11056740922796947603242351497, −21.04288842022304599150864503998, −20.329587864882433795686214360090, −19.721668020187775926319102574, −18.79574233337419274330437171639, −17.97532941612017975712118001702, −17.02413364296642010371371495427, −16.282563964316280287867135485767, −15.43488255567520227089044448340, −14.57929862663208674681199883865, −13.84921526798971697077792583003, −12.77255790334721910801005271185, −11.83975774519875248672361102081, −11.284660018456866851125108470679, −10.21963646646966047110006858283, −9.4100205850869796351674334487, −8.15447317658373219376696316644, −7.40862894932241022910774981767, −6.94232883659637605371788149417, −5.292439739540595067208200014086, −4.4777935807393651546237541704, −3.74057668402912282130308832477, −2.44386043297106204182071851144, −1.11925479913938727302822299508,
0.54254144494500601584163300266, 2.20460211001978997794446437296, 3.18656122430866331296896117418, 4.194839605840139049679553862634, 5.322559198550801196533732842135, 6.01782921361614689021080952688, 7.4338898820333930986611520786, 8.18035441070530046010533284970, 8.679627302743322084074515266027, 10.06200849514213228064082593401, 10.92682674455898221640510273659, 11.74402648217242112828950282838, 12.46019915872104190807484585900, 13.299845982639128533980305264941, 14.680920223876984711982792009668, 15.037657584956292489598320573937, 15.93565106418439013651443545458, 16.72813659330034848594881732514, 17.75011100173345290245125416698, 18.79138676161619248853458545294, 19.03818077160770162533823430127, 20.23269944970304385606239190777, 20.84293157990243440283238763560, 21.957687732610802108384997106368, 22.40340238084872137329399475403