| L(s) = 1 | + 2·5-s − 2·9-s − 4·13-s − 4·17-s + 3·25-s − 2·29-s − 4·37-s + 2·41-s − 4·45-s + 49-s − 2·53-s − 8·65-s + 2·73-s + 81-s − 8·85-s + 4·89-s + 2·97-s − 2·109-s − 2·113-s + 8·117-s − 2·121-s + 6·125-s + 127-s + 131-s + 137-s + 139-s − 4·145-s + ⋯ |
| L(s) = 1 | + 2·5-s − 2·9-s − 4·13-s − 4·17-s + 3·25-s − 2·29-s − 4·37-s + 2·41-s − 4·45-s + 49-s − 2·53-s − 8·65-s + 2·73-s + 81-s − 8·85-s + 4·89-s + 2·97-s − 2·109-s − 2·113-s + 8·117-s − 2·121-s + 6·125-s + 127-s + 131-s + 137-s + 139-s − 4·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1131433957\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1131433957\) |
| \(L(1)\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 13 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 3 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 31 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 37 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 47 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 71 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 79 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
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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
−6.35337070025047629932151857339, −6.24991109110888364397112315840, −6.07851483227756286233927865768, −5.64011634862853975186770794091, −5.59400436789861822833214036693, −5.36216480353331899338149464056, −5.12238917039444003323845612624, −5.09463039508427394149942211589, −4.79813089127332267040309986254, −4.66992909692582957122641413790, −4.64188579556747887412112291316, −4.15710256896083992987835924500, −3.87170765802163947924408705116, −3.72297973228272409604362185610, −3.29418406477466937147718645681, −3.07925424895727636799745837694, −2.72514058039281646679440132340, −2.58701077916360124221024802476, −2.49472955602083899395348013830, −2.17726947619573286590895958363, −1.93855197374750117449223587516, −1.90094108059558862213234290776, −1.77293994692815990411779655444, −0.824357360956548626325603439235, −0.14011205153890406935379395179,
0.14011205153890406935379395179, 0.824357360956548626325603439235, 1.77293994692815990411779655444, 1.90094108059558862213234290776, 1.93855197374750117449223587516, 2.17726947619573286590895958363, 2.49472955602083899395348013830, 2.58701077916360124221024802476, 2.72514058039281646679440132340, 3.07925424895727636799745837694, 3.29418406477466937147718645681, 3.72297973228272409604362185610, 3.87170765802163947924408705116, 4.15710256896083992987835924500, 4.64188579556747887412112291316, 4.66992909692582957122641413790, 4.79813089127332267040309986254, 5.09463039508427394149942211589, 5.12238917039444003323845612624, 5.36216480353331899338149464056, 5.59400436789861822833214036693, 5.64011634862853975186770794091, 6.07851483227756286233927865768, 6.24991109110888364397112315840, 6.35337070025047629932151857339
