| L(s) = 1 | + 4-s + 4·5-s + 9-s − 4·11-s + 4·20-s + 5·25-s + 16·31-s + 36-s + 8·41-s − 4·44-s + 4·45-s − 16·55-s + 20·59-s − 4·61-s − 64-s + 48·71-s − 20·89-s − 4·99-s + 5·100-s + 16·101-s + 20·109-s + 26·121-s + 16·124-s − 4·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1.78·5-s + 1/3·9-s − 1.20·11-s + 0.894·20-s + 25-s + 2.87·31-s + 1/6·36-s + 1.24·41-s − 0.603·44-s + 0.596·45-s − 2.15·55-s + 2.60·59-s − 0.512·61-s − 1/8·64-s + 5.69·71-s − 2.11·89-s − 0.402·99-s + 1/2·100-s + 1.59·101-s + 1.91·109-s + 2.36·121-s + 1.43·124-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.010026459\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.010026459\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 7 | | \( 1 \) |
| good | 11 | $C_2^2$ | \( ( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \) |
| 19 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 30 T^{2} + 371 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 12 T + 107 T^{2} - 12 p T^{3} + p^{2} T^{4} )( 1 + 12 T + 107 T^{2} + 12 p T^{3} + p^{2} T^{4} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 30 T^{2} - 1309 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 70 T^{2} + 2091 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 10 T + 41 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 70 T^{2} + 411 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 + 130 T^{2} + 11571 T^{4} + 130 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 150 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 10 T + 11 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2} \) |
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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
−6.93743255282117234196903430238, −6.37391262335333477219694472631, −6.29965613699331467012539237489, −6.27470747531857711755662236021, −5.92345023724382359239524440809, −5.61965115653977744326222805700, −5.47069668255571452869749068466, −5.35217249131181093973326512021, −5.28256237295560185596622820414, −4.61656698214323637008634901469, −4.51442523509593169177359648808, −4.47961252043037423832187204589, −4.38046525562699217048948108261, −3.59036905987497069791817264788, −3.58202910004163990823283575646, −3.22054808321400739463072999434, −3.12026566243053883354757650983, −2.51853106904935316420383406860, −2.39121487873036968390587065416, −2.22808566479561012462342435614, −2.10481797138180298725424723454, −1.79178703458543014239233849718, −1.03931858519793552282828813469, −0.974165179803274292867103777158, −0.57888197889722498363021273807,
0.57888197889722498363021273807, 0.974165179803274292867103777158, 1.03931858519793552282828813469, 1.79178703458543014239233849718, 2.10481797138180298725424723454, 2.22808566479561012462342435614, 2.39121487873036968390587065416, 2.51853106904935316420383406860, 3.12026566243053883354757650983, 3.22054808321400739463072999434, 3.58202910004163990823283575646, 3.59036905987497069791817264788, 4.38046525562699217048948108261, 4.47961252043037423832187204589, 4.51442523509593169177359648808, 4.61656698214323637008634901469, 5.28256237295560185596622820414, 5.35217249131181093973326512021, 5.47069668255571452869749068466, 5.61965115653977744326222805700, 5.92345023724382359239524440809, 6.27470747531857711755662236021, 6.29965613699331467012539237489, 6.37391262335333477219694472631, 6.93743255282117234196903430238
