| L(s) = 1 | + 4·7-s − 2·9-s + 4·17-s + 20·23-s + 12·25-s − 8·31-s − 4·41-s + 10·49-s − 8·63-s − 36·71-s + 48·73-s + 8·79-s + 3·81-s + 20·89-s − 32·97-s − 24·103-s + 16·119-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 8·153-s + 157-s + 80·161-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 2/3·9-s + 0.970·17-s + 4.17·23-s + 12/5·25-s − 1.43·31-s − 0.624·41-s + 10/7·49-s − 1.00·63-s − 4.27·71-s + 5.61·73-s + 0.900·79-s + 1/3·81-s + 2.11·89-s − 3.24·97-s − 2.36·103-s + 1.46·119-s + 1.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 0.646·153-s + 0.0798·157-s + 6.30·161-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.552940897\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.552940897\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 12 T^{2} + 74 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 234 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 246 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 2 T + 8 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 54 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 10 T + 68 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 37 | $D_4\times C_2$ | \( 1 - 44 T^{2} + 2454 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 116 T^{2} + 6294 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 12 T^{2} - 5290 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 18 T + 220 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 24 T + 278 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 300 T^{2} + 36086 T^{4} - 300 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 10 T + 200 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 16 T + 246 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.69553570883264387635495102050, −7.13855907419891903546172382423, −7.03589644736988787639717793448, −7.01969073645720311612967844410, −6.76621368751419490871232232448, −6.51979976680472075725341591515, −6.13161266203420683218625201564, −5.69279163159107495288387391420, −5.58411069973399102272998092884, −5.37868642932451268130022450942, −5.01238559126015744436922975078, −4.91338545921509558595985555953, −4.91145292875108467442409102525, −4.44171180471367354807683454683, −4.23682197105787509626874587641, −3.60952055061234962298056007745, −3.50704449651644137770354065918, −3.18083657379653966542518457480, −2.92734831132646356072402768093, −2.70585151279635278930060105283, −2.27038949755171512293374957717, −1.83789826925857281398348692097, −1.34675556196981808255592338859, −1.06202795561126622743477763683, −0.72040254140763017645693044492,
0.72040254140763017645693044492, 1.06202795561126622743477763683, 1.34675556196981808255592338859, 1.83789826925857281398348692097, 2.27038949755171512293374957717, 2.70585151279635278930060105283, 2.92734831132646356072402768093, 3.18083657379653966542518457480, 3.50704449651644137770354065918, 3.60952055061234962298056007745, 4.23682197105787509626874587641, 4.44171180471367354807683454683, 4.91145292875108467442409102525, 4.91338545921509558595985555953, 5.01238559126015744436922975078, 5.37868642932451268130022450942, 5.58411069973399102272998092884, 5.69279163159107495288387391420, 6.13161266203420683218625201564, 6.51979976680472075725341591515, 6.76621368751419490871232232448, 7.01969073645720311612967844410, 7.03589644736988787639717793448, 7.13855907419891903546172382423, 7.69553570883264387635495102050
