| L(s) = 1 | + 4·2-s + 10·4-s + 20·8-s + 35·16-s − 24·23-s + 56·32-s − 96·46-s + 10·49-s − 24·53-s + 84·64-s + 40·79-s − 9·81-s − 240·92-s + 40·98-s − 96·106-s − 48·107-s − 40·109-s − 24·113-s + 44·121-s + 127-s + 120·128-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 5·4-s + 7.07·8-s + 35/4·16-s − 5.00·23-s + 9.89·32-s − 14.1·46-s + 10/7·49-s − 3.29·53-s + 21/2·64-s + 4.50·79-s − 81-s − 25.0·92-s + 4.04·98-s − 9.32·106-s − 4.64·107-s − 3.83·109-s − 2.25·113-s + 4·121-s + 0.0887·127-s + 10.6·128-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.83945371\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.83945371\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| good | 11 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 160 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 + 170 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.87736987706637292530424011149, −6.59195397511434724454294308115, −6.39826258847772220156967743182, −6.32784738049015520781449809598, −6.19599517255180065643941997606, −6.05666015738576036634297062126, −5.66557650030998166521366179812, −5.34440495652606463811328999954, −5.29220959731045370681849633635, −5.10737014102720669465489711872, −4.83383220856005023640109665178, −4.44914481131758614782842023879, −4.22092591541903285299094398198, −4.00306465912384815511414852842, −3.83986375980348119262798929270, −3.69425169405336512398475280866, −3.61985538414152557382987293428, −2.86739255460033255843117141216, −2.80712017593564731998153868036, −2.54215467349350583926367200452, −2.27452239723119993113642430341, −1.80267881192866430480556060680, −1.65756770968313492172762065628, −1.38788309475886548449170190107, −0.35018346296517068620400736982,
0.35018346296517068620400736982, 1.38788309475886548449170190107, 1.65756770968313492172762065628, 1.80267881192866430480556060680, 2.27452239723119993113642430341, 2.54215467349350583926367200452, 2.80712017593564731998153868036, 2.86739255460033255843117141216, 3.61985538414152557382987293428, 3.69425169405336512398475280866, 3.83986375980348119262798929270, 4.00306465912384815511414852842, 4.22092591541903285299094398198, 4.44914481131758614782842023879, 4.83383220856005023640109665178, 5.10737014102720669465489711872, 5.29220959731045370681849633635, 5.34440495652606463811328999954, 5.66557650030998166521366179812, 6.05666015738576036634297062126, 6.19599517255180065643941997606, 6.32784738049015520781449809598, 6.39826258847772220156967743182, 6.59195397511434724454294308115, 6.87736987706637292530424011149
