| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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
−11.80654260350350639104803027145, −11.66204482762351348452938437915, −10.70675121732339552980590168191, −10.46757056628613997096260562229, −9.845488696155923613495161358816, −9.464125373831787409477369094998, −8.545308680119485617164219710243, −8.168025775250214814305200336901, −7.86510801978949900108947566195, −6.94626605041548264089391234487, −6.18274542521633997434313970606, −5.89743731398517553362918009104, −5.17396498807482459774869503876, −4.34293462415578763491945557323, −3.33461614675971696163153535013, −3.20954408712423141471276345607, −2.11823385541529878431332607138, −1.47624211095597730836290996338, 0, 0,
1.47624211095597730836290996338, 2.11823385541529878431332607138, 3.20954408712423141471276345607, 3.33461614675971696163153535013, 4.34293462415578763491945557323, 5.17396498807482459774869503876, 5.89743731398517553362918009104, 6.18274542521633997434313970606, 6.94626605041548264089391234487, 7.86510801978949900108947566195, 8.168025775250214814305200336901, 8.545308680119485617164219710243, 9.464125373831787409477369094998, 9.845488696155923613495161358816, 10.46757056628613997096260562229, 10.70675121732339552980590168191, 11.66204482762351348452938437915, 11.80654260350350639104803027145
