| L(s) = 1 | − 20·3-s − 250·5-s + 100·7-s + 790·9-s − 4.54e3·11-s + 3.54e3·13-s + 5.00e3·15-s − 2.73e4·17-s − 3.87e4·19-s − 2.00e3·21-s + 1.24e5·23-s + 4.68e4·25-s − 6.73e4·27-s − 7.22e4·29-s − 3.06e5·31-s + 9.08e4·33-s − 2.50e4·35-s − 1.23e5·37-s − 7.08e4·39-s + 2.64e5·41-s − 4.23e5·43-s − 1.97e5·45-s + 1.05e5·47-s − 1.40e6·49-s + 5.46e5·51-s − 2.39e6·53-s + 1.13e6·55-s + ⋯ |
| L(s) = 1 | − 0.427·3-s − 0.894·5-s + 0.110·7-s + 0.361·9-s − 1.02·11-s + 0.446·13-s + 0.382·15-s − 1.34·17-s − 1.29·19-s − 0.0471·21-s + 2.12·23-s + 3/5·25-s − 0.658·27-s − 0.550·29-s − 1.84·31-s + 0.440·33-s − 0.0985·35-s − 0.399·37-s − 0.191·39-s + 0.599·41-s − 0.811·43-s − 0.323·45-s + 0.148·47-s − 1.70·49-s + 0.577·51-s − 2.20·53-s + 0.920·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p^{3} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 20 T - 130 p T^{2} + 20 p^{7} T^{3} + p^{14} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 100 T + 1411250 T^{2} - 100 p^{7} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4544 T + 31976326 T^{2} + 4544 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 3540 T + 100535470 T^{2} - 3540 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 27340 T + 901005190 T^{2} + 27340 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2040 p T + 113449762 p T^{2} + 2040 p^{8} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 124140 T + 10649684530 T^{2} - 124140 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 72260 T + 6846819118 T^{2} + 72260 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 306824 T + 77964629966 T^{2} + 306824 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 123020 T + 144088599870 T^{2} + 123020 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 264364 T + 161786388886 T^{2} - 264364 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 423300 T + 446651231050 T^{2} + 423300 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 105460 T + 858715356610 T^{2} - 105460 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2391580 T + 3562552504510 T^{2} + 2391580 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 1120120 T + 1362334883638 T^{2} - 1120120 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 2257044 T + 5613447576526 T^{2} - 2257044 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4516460 T + 16742087664890 T^{2} + 4516460 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 621784 T + 17914494152446 T^{2} + 621784 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4569060 T + 23424949855030 T^{2} - 4569060 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 4333040 T + 330830231042 p T^{2} + 4333040 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 9793020 T + 59971104320890 T^{2} - 9793020 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 6025620 T + 89865866149558 T^{2} - 6025620 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4609540 T + 142930351581510 T^{2} - 4609540 p^{7} T^{3} + p^{14} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.83549682500989205603644636814, −12.21967363351398099177606413264, −11.22937931632661916493036518577, −11.03936569705043154232338556055, −10.90546839166807490708100419847, −10.00834977877143786542370829783, −9.084107421320057133754728803736, −8.825377046093575334798563901795, −8.001532631737234957529930191733, −7.49806566903736678997121414049, −6.80270889511309676160153629985, −6.31400098865013419386378320927, −5.22824689648290207417389627842, −4.85772134898464350070788117204, −4.02593693912856197711876505528, −3.31092972516631335195459187649, −2.31282382187266457660712532502, −1.39341613178606068184573481017, 0, 0,
1.39341613178606068184573481017, 2.31282382187266457660712532502, 3.31092972516631335195459187649, 4.02593693912856197711876505528, 4.85772134898464350070788117204, 5.22824689648290207417389627842, 6.31400098865013419386378320927, 6.80270889511309676160153629985, 7.49806566903736678997121414049, 8.001532631737234957529930191733, 8.825377046093575334798563901795, 9.084107421320057133754728803736, 10.00834977877143786542370829783, 10.90546839166807490708100419847, 11.03936569705043154232338556055, 11.22937931632661916493036518577, 12.21967363351398099177606413264, 12.83549682500989205603644636814
