| L(s) = 1 | + 14·3-s − 42·5-s + 686·7-s − 698·9-s − 7.42e3·11-s − 1.18e4·13-s − 588·15-s + 1.57e4·17-s − 2.66e4·19-s + 9.60e3·21-s + 3.26e4·23-s − 1.23e5·25-s + 8.33e3·27-s + 1.58e5·29-s − 1.80e5·31-s − 1.03e5·33-s − 2.88e4·35-s + 4.58e4·37-s − 1.65e5·39-s − 3.21e5·41-s − 1.02e6·43-s + 2.93e4·45-s + 1.66e6·47-s + 3.52e5·49-s + 2.21e5·51-s + 4.10e5·53-s + 3.11e5·55-s + ⋯ |
| L(s) = 1 | + 0.299·3-s − 0.150·5-s + 0.755·7-s − 0.319·9-s − 1.68·11-s − 1.49·13-s − 0.0449·15-s + 0.779·17-s − 0.890·19-s + 0.226·21-s + 0.559·23-s − 1.57·25-s + 0.0814·27-s + 1.20·29-s − 1.08·31-s − 0.503·33-s − 0.113·35-s + 0.148·37-s − 0.447·39-s − 0.729·41-s − 1.96·43-s + 0.0479·45-s + 2.34·47-s + 3/7·49-s + 0.233·51-s + 0.378·53-s + 0.252·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 14 T + 298 p T^{2} - 14 p^{7} T^{3} + p^{14} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 42 T + 24986 p T^{2} + 42 p^{7} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 7428 T + 46542982 T^{2} + 7428 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 70 p^{2} T + 160198410 T^{2} + 70 p^{9} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 15792 T + 526157566 T^{2} - 15792 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 26614 T + 1962247086 T^{2} + 26614 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 32640 T - 991808882 T^{2} - 32640 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 158016 T + 39988772806 T^{2} - 158016 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 180740 T + 38484320958 T^{2} + 180740 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 45824 T - 18085535274 T^{2} - 45824 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 321720 T + 181439440606 T^{2} + 321720 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 1023868 T + 671194246566 T^{2} + 1023868 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 1665972 T + 1675477834078 T^{2} - 1665972 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 410628 T + 2334205080574 T^{2} - 410628 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 1702134 T + 4026188129518 T^{2} + 1702134 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 547526 T + 5609323300002 T^{2} - 547526 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2590616 T + 4654840470246 T^{2} - 2590616 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4129272 T + 22218672158062 T^{2} - 4129272 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8008868 T + 520866105078 p T^{2} + 8008868 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2470456 T - 13654752819234 T^{2} - 2470456 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 9900786 T + 68835957963214 T^{2} - 9900786 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 15423492 T + 143317325773078 T^{2} - 15423492 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 17377472 T + 164552259333822 T^{2} + 17377472 p^{7} T^{3} + p^{14} T^{4} \) |
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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
−9.792344650203084942746359083932, −9.268394343072967787370365936121, −8.679667518108256458971007937448, −8.289444645509624351905260963871, −7.74326505353381115431312212636, −7.72948972812547488536237981551, −7.05527979620062080606284663063, −6.58881186931987587118151143591, −5.63791751515881964954358531708, −5.54735735472058596774062131151, −4.84613614095476282097981693385, −4.67451667577828574514664907584, −3.82636698045535018116626317221, −3.29992537376788919285413819443, −2.47324528288732912192267168861, −2.42365064276245658957870385685, −1.72771849518347182965807401129, −0.911263707181631721031826494789, 0, 0,
0.911263707181631721031826494789, 1.72771849518347182965807401129, 2.42365064276245658957870385685, 2.47324528288732912192267168861, 3.29992537376788919285413819443, 3.82636698045535018116626317221, 4.67451667577828574514664907584, 4.84613614095476282097981693385, 5.54735735472058596774062131151, 5.63791751515881964954358531708, 6.58881186931987587118151143591, 7.05527979620062080606284663063, 7.72948972812547488536237981551, 7.74326505353381115431312212636, 8.289444645509624351905260963871, 8.679667518108256458971007937448, 9.268394343072967787370365936121, 9.792344650203084942746359083932
