| L(s) = 1 | − 6·2-s + 9·4-s + 30·8-s − 140·16-s − 12·17-s − 76·19-s + 60·23-s + 36·31-s + 126·32-s + 72·34-s + 456·38-s − 360·46-s − 48·47-s + 88·49-s + 204·53-s + 76·61-s − 216·62-s + 459·64-s − 108·68-s − 684·76-s + 300·79-s + 36·83-s + 540·92-s + 288·94-s − 528·98-s − 1.22e3·106-s + 24·107-s + ⋯ |
| L(s) = 1 | − 3·2-s + 9/4·4-s + 15/4·8-s − 8.75·16-s − 0.705·17-s − 4·19-s + 2.60·23-s + 1.16·31-s + 3.93·32-s + 2.11·34-s + 12·38-s − 7.82·46-s − 1.02·47-s + 1.79·49-s + 3.84·53-s + 1.24·61-s − 3.48·62-s + 7.17·64-s − 1.58·68-s − 9·76-s + 3.79·79-s + 0.433·83-s + 5.86·92-s + 3.06·94-s − 5.38·98-s − 11.5·106-s + 0.224·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3527358030\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3527358030\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( ( 1 + 3 T + 9 T^{2} + 3 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 88 T^{2} + 5118 T^{4} - 88 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 216 T^{2} + 36446 T^{4} - 216 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 584 T^{2} + 142206 T^{4} - 584 p^{4} T^{6} + p^{8} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 6 T + 507 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 2 p T + 903 T^{2} + 2 p^{3} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 - 30 T + 1103 T^{2} - 30 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 216 T^{2} - 1234 p^{2} T^{4} - 216 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 18 T + 383 T^{2} - 18 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 1916 T^{2} + 3514086 T^{4} - 1916 p^{4} T^{6} + p^{8} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 1224 T^{2} + 5946686 T^{4} + 1224 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 136 T^{2} + 5078046 T^{4} + 136 p^{4} T^{6} + p^{8} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 24 T + 1182 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 - 102 T + 7719 T^{2} - 102 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 3384 T^{2} + 26924606 T^{4} + 3384 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $C_2$ | \( ( 1 - 19 T + p^{2} T^{2} )^{4} \) |
| 67 | $D_4\times C_2$ | \( 1 - 6364 T^{2} + 16835046 T^{4} - 6364 p^{4} T^{6} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 7896 T^{2} + 66189566 T^{4} - 7896 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 9368 T^{2} + 67214718 T^{4} - 9368 p^{4} T^{6} + p^{8} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 150 T + 17387 T^{2} - 150 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 - 18 T + 10979 T^{2} - 18 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 23656 T^{2} + 264671646 T^{4} - 23656 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 5356 T^{2} + 168882726 T^{4} + 5356 p^{4} T^{6} + p^{8} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.50962050638129073451947341382, −7.13190193842903140804733951040, −7.00670691320117785594866374885, −6.78356544654656688905114078559, −6.63521717735912869021892917894, −6.52196851961997051563699180485, −5.88636104006275099049156315005, −5.74561786195739008089958527295, −5.45254352102942392684205511720, −5.07740470677467240497964890959, −4.72846113362431309110236555415, −4.67965801789320651249209930795, −4.48064371440739818998523779257, −4.08388660950456453096317425166, −3.91430276294251798861329251723, −3.80361416226858691100775212144, −3.11959650746473241301572570864, −2.79542670605320430831441066188, −2.19188685896074619933147380002, −2.08857353074659077167218940236, −1.96296581437555210386049974862, −1.02369648689540047246564834341, −1.00676078411075651260891771537, −0.49777516592633552374562985781, −0.41676208437313576133459916042,
0.41676208437313576133459916042, 0.49777516592633552374562985781, 1.00676078411075651260891771537, 1.02369648689540047246564834341, 1.96296581437555210386049974862, 2.08857353074659077167218940236, 2.19188685896074619933147380002, 2.79542670605320430831441066188, 3.11959650746473241301572570864, 3.80361416226858691100775212144, 3.91430276294251798861329251723, 4.08388660950456453096317425166, 4.48064371440739818998523779257, 4.67965801789320651249209930795, 4.72846113362431309110236555415, 5.07740470677467240497964890959, 5.45254352102942392684205511720, 5.74561786195739008089958527295, 5.88636104006275099049156315005, 6.52196851961997051563699180485, 6.63521717735912869021892917894, 6.78356544654656688905114078559, 7.00670691320117785594866374885, 7.13190193842903140804733951040, 7.50962050638129073451947341382
