| L(s) = 1 | − 4·4-s + 4·16-s + 168·19-s − 160·31-s − 146·49-s + 388·61-s + 16·64-s − 672·76-s + 468·79-s + 64·109-s − 480·121-s + 640·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 514·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 4-s + 1/4·16-s + 8.84·19-s − 5.16·31-s − 2.97·49-s + 6.36·61-s + 1/4·64-s − 8.84·76-s + 5.92·79-s + 0.587·109-s − 3.96·121-s + 5.16·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 3.04·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(12.00869095\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.00869095\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 3 | \( 1 \) |
| 5 | \( 1 - p^{4} T^{4} + p^{8} T^{8} \) |
| good | 7 | \( ( 1 + 73 T^{2} + 2928 T^{4} + 73 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 11 | \( ( 1 + 240 T^{2} + 42959 T^{4} + 240 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 13 | \( ( 1 + 257 T^{2} + 37488 T^{4} + 257 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 17 | \( ( 1 + 450 T^{2} + p^{4} T^{4} )^{4} \) |
| 19 | \( ( 1 - 21 T + p^{2} T^{2} )^{8} \) |
| 23 | \( ( 1 - 1056 T^{2} + 835295 T^{4} - 1056 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 29 | \( ( 1 + 224 T^{2} - 657105 T^{4} + 224 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 31 | \( ( 1 + 40 T + 639 T^{2} + 40 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 37 | \( ( 1 - 2113 T^{2} + p^{4} T^{4} )^{4} \) |
| 41 | \( ( 1 + 624 T^{2} - 2436385 T^{4} + 624 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 43 | \( ( 1 - 398 T^{2} - 3260397 T^{4} - 398 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 47 | \( ( 1 - 3906 T^{2} + 10377155 T^{4} - 3906 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 53 | \( ( 1 + 416 T^{2} + p^{4} T^{4} )^{4} \) |
| 59 | \( ( 1 - 1230 T^{2} - 10604461 T^{4} - 1230 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 61 | \( ( 1 - 97 T + 5688 T^{2} - 97 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 67 | \( ( 1 - 8183 T^{2} + 46810368 T^{4} - 8183 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 71 | \( ( 1 - 2144 T^{2} + p^{4} T^{4} )^{4} \) |
| 73 | \( ( 1 - 10369 T^{2} + p^{4} T^{4} )^{4} \) |
| 79 | \( ( 1 - 117 T + 7448 T^{2} - 117 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 83 | \( ( 1 - 10416 T^{2} + 61034735 T^{4} - 10416 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 89 | \( ( 1 + 5790 T^{2} + p^{4} T^{4} )^{4} \) |
| 97 | \( ( 1 + 17137 T^{2} + 205147488 T^{4} + 17137 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.02324530294607785293261822709, −3.83188437477666988648593818264, −3.80922699065105870693553932841, −3.73097674905561953973864622288, −3.59069449348584302374247285746, −3.48646593671526344318068482886, −3.42650671714225509604573880666, −3.34880818431540144540785073698, −3.28643877428678392925611519603, −3.11648725567065166246980110760, −2.86001927329270523351376435066, −2.52691125427826533948238085525, −2.48704269522586107617181502057, −2.30366631539596373474114719061, −2.19793182030713951831146178494, −2.12040189760577118135615466054, −1.57788808176341063957022886781, −1.56289154390385078560417499282, −1.27972062365653918585655501789, −1.25410774572207089497023373181, −1.06215654488537593490921724763, −0.972108097046276455999098549905, −0.48452217071774126974236545007, −0.47682316074921286169173281240, −0.32299818180651178731216564562,
0.32299818180651178731216564562, 0.47682316074921286169173281240, 0.48452217071774126974236545007, 0.972108097046276455999098549905, 1.06215654488537593490921724763, 1.25410774572207089497023373181, 1.27972062365653918585655501789, 1.56289154390385078560417499282, 1.57788808176341063957022886781, 2.12040189760577118135615466054, 2.19793182030713951831146178494, 2.30366631539596373474114719061, 2.48704269522586107617181502057, 2.52691125427826533948238085525, 2.86001927329270523351376435066, 3.11648725567065166246980110760, 3.28643877428678392925611519603, 3.34880818431540144540785073698, 3.42650671714225509604573880666, 3.48646593671526344318068482886, 3.59069449348584302374247285746, 3.73097674905561953973864622288, 3.80922699065105870693553932841, 3.83188437477666988648593818264, 4.02324530294607785293261822709
Plot not available for L-functions of degree greater than 10.
