L(s) = 1 | + (−0.580 + 0.814i)2-s + (0.5 − 0.866i)3-s + (−0.327 − 0.945i)4-s + (−0.723 − 0.690i)5-s + (0.415 + 0.909i)6-s + (0.959 + 0.281i)8-s + (−0.5 − 0.866i)9-s + (0.981 − 0.189i)10-s + (−0.981 − 0.189i)12-s + (−0.654 + 0.755i)13-s + (−0.959 + 0.281i)15-s + (−0.786 + 0.618i)16-s + (0.0475 + 0.998i)17-s + (0.995 + 0.0950i)18-s + (0.0475 − 0.998i)19-s + (−0.415 + 0.909i)20-s + ⋯ |
L(s) = 1 | + (−0.580 + 0.814i)2-s + (0.5 − 0.866i)3-s + (−0.327 − 0.945i)4-s + (−0.723 − 0.690i)5-s + (0.415 + 0.909i)6-s + (0.959 + 0.281i)8-s + (−0.5 − 0.866i)9-s + (0.981 − 0.189i)10-s + (−0.981 − 0.189i)12-s + (−0.654 + 0.755i)13-s + (−0.959 + 0.281i)15-s + (−0.786 + 0.618i)16-s + (0.0475 + 0.998i)17-s + (0.995 + 0.0950i)18-s + (0.0475 − 0.998i)19-s + (−0.415 + 0.909i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.498 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.498 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1423088874 + 0.2461280564i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1423088874 + 0.2461280564i\) |
\(L(1)\) |
\(\approx\) |
\(0.6190973443 + 0.007820059223i\) |
\(L(1)\) |
\(\approx\) |
\(0.6190973443 + 0.007820059223i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.580 + 0.814i)T \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 + (-0.723 - 0.690i)T \) |
| 13 | \( 1 + (-0.654 + 0.755i)T \) |
| 17 | \( 1 + (0.0475 + 0.998i)T \) |
| 19 | \( 1 + (0.0475 - 0.998i)T \) |
| 23 | \( 1 + (-0.786 + 0.618i)T \) |
| 29 | \( 1 + (-0.841 - 0.540i)T \) |
| 31 | \( 1 + (-0.981 + 0.189i)T \) |
| 37 | \( 1 + (-0.327 + 0.945i)T \) |
| 41 | \( 1 + (0.415 + 0.909i)T \) |
| 43 | \( 1 + (0.959 + 0.281i)T \) |
| 47 | \( 1 + (0.995 - 0.0950i)T \) |
| 53 | \( 1 + (-0.786 - 0.618i)T \) |
| 59 | \( 1 + (-0.580 - 0.814i)T \) |
| 61 | \( 1 + (-0.995 + 0.0950i)T \) |
| 67 | \( 1 + (-0.995 - 0.0950i)T \) |
| 71 | \( 1 + (0.841 + 0.540i)T \) |
| 73 | \( 1 + (0.928 + 0.371i)T \) |
| 79 | \( 1 + (-0.723 - 0.690i)T \) |
| 83 | \( 1 + (-0.142 + 0.989i)T \) |
| 89 | \( 1 + (0.888 + 0.458i)T \) |
| 97 | \( 1 + (0.959 + 0.281i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−21.81930582863431117646792388460, −20.78419187770056965962841279481, −20.231720758177567935538785979017, −19.63703576597552944619855608756, −18.74000269193280267189484465449, −18.15773440074145412681176666545, −17.01657210870434898685712609531, −16.224372143020166728701729384513, −15.547185601023462228149396729709, −14.49091701794966441917142062247, −13.96240328624408023634653845739, −12.600044652807815025255147301488, −11.95039315622753977962421482975, −10.809267360861434657631242811776, −10.55259035030197287487215055670, −9.55110254448962745433330609147, −8.835220435264242258890198404585, −7.6698746076710992048458089876, −7.46018787329360183444845322306, −5.64666018596849801574999347709, −4.44314181655353256187156633463, −3.68937908145295516682439031179, −2.927136774127422102530641521998, −2.0953119325195728731504825342, −0.1530715377931523152836108091,
1.217444756716266148849359890341, 2.13507145798639916116518390824, 3.71873297339472556454998960243, 4.70198374763653036807589375918, 5.8078737481795787764803262374, 6.73942127998450129184671931129, 7.59729907275856452819009922701, 8.07253581746257587388138123562, 9.06466815033561095393346593610, 9.51123775332140773608385605812, 10.98028577160596190484528522002, 11.85323415190331063385134589260, 12.78953410649988918392329803763, 13.54723593860128288303874354456, 14.47151781116524702993806725136, 15.17278717503271012297396209995, 15.93960737148892873577646028122, 16.97587521432863459602100459172, 17.41298300835867876578043554856, 18.47448691575682143093076273875, 19.18638086040065204498628820881, 19.74130440126291188407776603927, 20.307317703890732087352701971496, 21.57470948885183540952287545829, 22.70302126166446528731774921821