| 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
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
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
−10.14759654096126172292054900684, −9.906735423876559944522278888663, −9.392509533904235749903020870755, −8.744250403730522029965301832825, −8.309654066339426997068781095106, −7.85516884770904436262469805530, −7.33518752232754479944588388458, −6.47494889560488943138541192753, −6.29635927099359631602123327751, −6.04349304265214704585510336314, −5.07287594282705500431697798248, −4.85719523664880921001178748757, −4.24325305187929678293016854091, −3.66685494644891322328796016540, −2.47109807778535457829067083657, −2.46752967321370260197207537944, −1.84151550449490971109143927663, −0.997161230813136600883221050474, 0, 0,
0.997161230813136600883221050474, 1.84151550449490971109143927663, 2.46752967321370260197207537944, 2.47109807778535457829067083657, 3.66685494644891322328796016540, 4.24325305187929678293016854091, 4.85719523664880921001178748757, 5.07287594282705500431697798248, 6.04349304265214704585510336314, 6.29635927099359631602123327751, 6.47494889560488943138541192753, 7.33518752232754479944588388458, 7.85516884770904436262469805530, 8.309654066339426997068781095106, 8.744250403730522029965301832825, 9.392509533904235749903020870755, 9.906735423876559944522278888663, 10.14759654096126172292054900684
