| L(s) = 1 | − 2·25-s − 8·29-s − 2·81-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯ |
| L(s) = 1 | − 2·25-s − 8·29-s − 2·81-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.06032388029\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06032388029\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + T^{4} \) |
| good | 3 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 29 | $C_1$ | \( ( 1 + T )^{8} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
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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
−6.62931597850986804223674062900, −6.28011466778037113510506983890, −6.14641907453021981221506252895, −5.83920138823794587695111952648, −5.79656272267362009738392787127, −5.66869594877539609422565847783, −5.40423386802856022619193045079, −5.24609300655022092760602989252, −5.01204313542183171913037913994, −4.89356266618034015865649900991, −4.20407433506505993895151178118, −4.15552479623073398087283616812, −4.05178209043252604440803154498, −3.93054745137848606669445419162, −3.51556376065953967518053034108, −3.42050594435959414388165862574, −3.35818709159905837849051285624, −2.78734616764231358974668198617, −2.47430116350105595534480659430, −2.17493756776900941870296590072, −1.99226898323628789385560294789, −1.84924287473662587634657924794, −1.36494030425422505290753424538, −1.34666811078089520755033665187, −0.10584117177015804727069645033,
0.10584117177015804727069645033, 1.34666811078089520755033665187, 1.36494030425422505290753424538, 1.84924287473662587634657924794, 1.99226898323628789385560294789, 2.17493756776900941870296590072, 2.47430116350105595534480659430, 2.78734616764231358974668198617, 3.35818709159905837849051285624, 3.42050594435959414388165862574, 3.51556376065953967518053034108, 3.93054745137848606669445419162, 4.05178209043252604440803154498, 4.15552479623073398087283616812, 4.20407433506505993895151178118, 4.89356266618034015865649900991, 5.01204313542183171913037913994, 5.24609300655022092760602989252, 5.40423386802856022619193045079, 5.66869594877539609422565847783, 5.79656272267362009738392787127, 5.83920138823794587695111952648, 6.14641907453021981221506252895, 6.28011466778037113510506983890, 6.62931597850986804223674062900
