| L(s) = 1 | − 20·7-s + 16·11-s − 40·17-s − 40·23-s + 16·31-s − 200·41-s − 120·43-s + 200·49-s − 200·53-s − 312·61-s + 40·67-s + 80·71-s + 20·73-s − 320·77-s − 240·83-s − 300·97-s − 232·101-s − 220·103-s − 160·107-s + 160·113-s + 800·119-s − 24·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + ⋯ |
| L(s) = 1 | − 2.85·7-s + 1.45·11-s − 2.35·17-s − 1.73·23-s + 0.516·31-s − 4.87·41-s − 2.79·43-s + 4.08·49-s − 3.77·53-s − 5.11·61-s + 0.597·67-s + 1.12·71-s + 0.273·73-s − 4.15·77-s − 2.89·83-s − 3.09·97-s − 2.29·101-s − 2.13·103-s − 1.49·107-s + 1.41·113-s + 6.72·119-s − 0.198·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.003231575987\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.003231575987\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $D_4\times C_2$ | \( 1 + 20 T + 200 T^{2} + 1740 T^{3} + 13694 T^{4} + 1740 p^{2} T^{5} + 200 p^{4} T^{6} + 20 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 8 T + 108 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^3$ | \( 1 - 55678 T^{4} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 40 T + 800 T^{2} + 19080 T^{3} + 419714 T^{4} + 19080 p^{2} T^{5} + 800 p^{4} T^{6} + 40 p^{6} T^{7} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 21126 T^{4} - 44 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 40 T + 800 T^{2} + 1080 p T^{3} + 1442 p^{2} T^{4} + 1080 p^{3} T^{5} + 800 p^{4} T^{6} + 40 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 2096 T^{2} + 2222466 T^{4} - 2096 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 8 T + 1338 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^3$ | \( 1 + 3168578 T^{4} + p^{8} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 100 T + 5262 T^{2} + 100 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 120 T + 7200 T^{2} + 432120 T^{3} + 22864898 T^{4} + 432120 p^{2} T^{5} + 7200 p^{4} T^{6} + 120 p^{6} T^{7} + p^{8} T^{8} \) |
| 47 | $C_2^3$ | \( 1 + 2115554 T^{4} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 200 T + 20000 T^{2} + 1475400 T^{3} + 87973634 T^{4} + 1475400 p^{2} T^{5} + 20000 p^{4} T^{6} + 200 p^{6} T^{7} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 5816 T^{2} + 18089586 T^{4} - 5816 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 156 T + 12926 T^{2} + 156 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 40 T + 800 T^{2} - 170280 T^{3} + 36190274 T^{4} - 170280 p^{2} T^{5} + 800 p^{4} T^{6} - 40 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 40 T + 882 T^{2} - 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 20 T + 200 T^{2} - 46140 T^{3} + 1512014 T^{4} - 46140 p^{2} T^{5} + 200 p^{4} T^{6} - 20 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 22612 T^{2} + 204343398 T^{4} - 22612 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 240 T + 28800 T^{2} + 2549040 T^{3} + 211683458 T^{4} + 2549040 p^{2} T^{5} + 28800 p^{4} T^{6} + 240 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 11716 T^{2} + 151160646 T^{4} + 11716 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 300 T + 45000 T^{2} + 5967300 T^{3} + 681431438 T^{4} + 5967300 p^{2} T^{5} + 45000 p^{4} T^{6} + 300 p^{6} T^{7} + p^{8} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.79711186214701889071575957603, −6.78785968167791407793687201335, −6.70081333151982088707957139009, −6.36089518871499768962806060150, −6.21175864535319024808327237093, −6.15575003248726827901780024662, −5.94001704414492331329523462833, −5.31968771731182756484833953663, −5.07268754444521176403891364045, −5.06202332828801916293171697672, −4.74065935577746486067368757365, −4.20991562786310433115056053718, −3.99792969105739034832541578476, −3.99709505015392392678793298047, −3.81927001463765195162686377481, −3.07211526923617230070337120269, −2.98934689583219295205237078154, −2.98123663213831171756239885986, −2.97471287909186233272616843940, −1.89841728828579633211552611419, −1.89806216474052496092223482801, −1.48131407859227937325094783715, −1.43141534876750915223718995505, −0.098341088069255370318731907616, −0.05827949042895518755581257476,
0.05827949042895518755581257476, 0.098341088069255370318731907616, 1.43141534876750915223718995505, 1.48131407859227937325094783715, 1.89806216474052496092223482801, 1.89841728828579633211552611419, 2.97471287909186233272616843940, 2.98123663213831171756239885986, 2.98934689583219295205237078154, 3.07211526923617230070337120269, 3.81927001463765195162686377481, 3.99709505015392392678793298047, 3.99792969105739034832541578476, 4.20991562786310433115056053718, 4.74065935577746486067368757365, 5.06202332828801916293171697672, 5.07268754444521176403891364045, 5.31968771731182756484833953663, 5.94001704414492331329523462833, 6.15575003248726827901780024662, 6.21175864535319024808327237093, 6.36089518871499768962806060150, 6.70081333151982088707957139009, 6.78785968167791407793687201335, 6.79711186214701889071575957603
