| L(s) = 1 | − 4·4-s − 4·5-s + 8·7-s + 26·9-s − 40·11-s + 12·16-s − 28·17-s + 16·20-s + 20·23-s − 90·25-s − 32·28-s − 32·35-s − 104·36-s − 76·43-s + 160·44-s − 104·45-s + 140·47-s − 108·49-s + 160·55-s − 148·61-s + 208·63-s − 32·64-s + 112·68-s − 196·73-s − 320·77-s − 48·80-s + 369·81-s + ⋯ |
| L(s) = 1 | − 4-s − 4/5·5-s + 8/7·7-s + 26/9·9-s − 3.63·11-s + 3/4·16-s − 1.64·17-s + 4/5·20-s + 0.869·23-s − 3.59·25-s − 8/7·28-s − 0.914·35-s − 2.88·36-s − 1.76·43-s + 3.63·44-s − 2.31·45-s + 2.97·47-s − 2.20·49-s + 2.90·55-s − 2.42·61-s + 3.30·63-s − 1/2·64-s + 1.64·68-s − 2.68·73-s − 4.15·77-s − 3/5·80-s + 41/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7175418936\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7175418936\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | | \( 1 \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 26 T^{2} + 307 T^{4} - 26 p^{4} T^{6} + p^{8} T^{8} \) |
| 5 | $C_2$ | \( ( 1 + T + p^{2} T^{2} )^{4} \) |
| 7 | $D_{4}$ | \( ( 1 - 4 T + 78 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 11 | $D_{4}$ | \( ( 1 + 20 T + 318 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 382 T^{2} + 72003 T^{4} - 382 p^{4} T^{6} + p^{8} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 14 T + 603 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 - 10 T + 597 T^{2} - 10 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 1342 T^{2} + 923907 T^{4} - 1342 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 - 950 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 4156 T^{2} + 8035302 T^{4} - 4156 p^{4} T^{6} + p^{8} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 386 T^{2} + 5307747 T^{4} + 386 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 38 T + 885 T^{2} + 38 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 - 70 T + 3477 T^{2} - 70 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 1726 T^{2} + 13816227 T^{4} - 1726 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 9274 T^{2} + 44455827 T^{4} - 9274 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 74 T + 7275 T^{2} + 74 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 346 T^{2} + 12803187 T^{4} - 346 p^{4} T^{6} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 4234 T^{2} + 50248707 T^{4} - 4234 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 98 T + 6915 T^{2} + 98 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 20554 T^{2} + 178849347 T^{4} - 20554 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 32 T + 13818 T^{2} + 32 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 31534 T^{2} + 374081907 T^{4} - 31534 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 1874 T^{2} + 172267827 T^{4} + 1874 p^{4} T^{6} + p^{8} T^{8} \) |
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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
−7.48268138577391442072908949248, −7.17560873093578800027824378347, −6.97064145008781681089415588150, −6.59138991880090852205105718477, −6.51609742379758545177083179368, −5.84290674813267589589922719877, −5.72048269293709133117201834305, −5.58590216623826268514161353615, −5.39911937220342951507239204231, −4.85986332375484930375101997917, −4.68157447173436949100346681069, −4.65361190657621217036803670537, −4.48892659030209703971524810850, −4.20406447630208634779163294602, −3.82325923405605806575322411178, −3.81238123437747049055135582912, −3.22019652850234187732438635556, −2.90197531166580453849065736813, −2.63547901292122259358084912359, −2.17957430998056915006776979044, −1.81457152382606896090577563121, −1.63494955449855115209803289089, −1.37343174383498841069561776729, −0.39541794064833733407490269052, −0.27107160892860222670693756454,
0.27107160892860222670693756454, 0.39541794064833733407490269052, 1.37343174383498841069561776729, 1.63494955449855115209803289089, 1.81457152382606896090577563121, 2.17957430998056915006776979044, 2.63547901292122259358084912359, 2.90197531166580453849065736813, 3.22019652850234187732438635556, 3.81238123437747049055135582912, 3.82325923405605806575322411178, 4.20406447630208634779163294602, 4.48892659030209703971524810850, 4.65361190657621217036803670537, 4.68157447173436949100346681069, 4.85986332375484930375101997917, 5.39911937220342951507239204231, 5.58590216623826268514161353615, 5.72048269293709133117201834305, 5.84290674813267589589922719877, 6.51609742379758545177083179368, 6.59138991880090852205105718477, 6.97064145008781681089415588150, 7.17560873093578800027824378347, 7.48268138577391442072908949248
