| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(6.623143710\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.623143710\) |
| \(L(\frac{9}{2})\) |
|
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
−10.71071835814020025401417101589, −10.07838082712966776604452604278, −9.455929994017889910986780564252, −9.390999402920456318269188466338, −8.861295988216719933406536709425, −8.567050995671673591853489475345, −7.72644037868311510892619685074, −7.12349748760849894396500468611, −6.95025370320674398485547437725, −6.43873406693991890102381684509, −5.67631901214199125210016672833, −5.30272985305392417952223606110, −4.62009435371070080739940169502, −4.25934533998519610304300559031, −3.37485491865883109273352361260, −2.94871852567240972318827718068, −2.29022043171348218955046464000, −1.70572765442910137810145057961, −1.09865931887500107871803583509, −0.58925247701350748620958762265,
0.58925247701350748620958762265, 1.09865931887500107871803583509, 1.70572765442910137810145057961, 2.29022043171348218955046464000, 2.94871852567240972318827718068, 3.37485491865883109273352361260, 4.25934533998519610304300559031, 4.62009435371070080739940169502, 5.30272985305392417952223606110, 5.67631901214199125210016672833, 6.43873406693991890102381684509, 6.95025370320674398485547437725, 7.12349748760849894396500468611, 7.72644037868311510892619685074, 8.567050995671673591853489475345, 8.861295988216719933406536709425, 9.390999402920456318269188466338, 9.455929994017889910986780564252, 10.07838082712966776604452604278, 10.71071835814020025401417101589
