| L(s) = 1 | + 8·7-s + 12·13-s − 8·25-s − 32·31-s − 12·37-s + 32·49-s + 32·67-s + 28·73-s + 96·91-s + 12·97-s − 8·103-s + 28·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 72·169-s + 173-s − 64·175-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 3.02·7-s + 3.32·13-s − 8/5·25-s − 5.74·31-s − 1.97·37-s + 32/7·49-s + 3.90·67-s + 3.27·73-s + 10.0·91-s + 1.21·97-s − 0.788·103-s + 2.54·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 5.53·169-s + 0.0760·173-s − 4.83·175-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.319379210\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.319379210\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 16 T^{2} + p^{2} T^{4} )( 1 + 16 T^{2} + p^{2} T^{4} ) \) |
| 19 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 734 T^{4} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 80 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 1918 T^{4} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 5582 T^{4} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 16 T + 128 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 83 | $C_2^3$ | \( 1 - 13294 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 176 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−9.322988967998302935069205618263, −8.740425684536063449337412965126, −8.630112055510370019449649098708, −8.538963608666549612557877313126, −8.436248783774728651012950058313, −7.76699232441108185938878281759, −7.73065782167575265624822082327, −7.58431261619051506237199756712, −7.24372312197739519344453805176, −6.76541269617338065495478811558, −6.50995405155104581071061562410, −6.09879207042107776860036845366, −5.82089910018840341214098844600, −5.34225973896219813567878461256, −5.33294689173421237124892140232, −5.09173433140503458662083098237, −4.78086893111247924514674194249, −3.80404894738793192763103248237, −3.80225388967424673482272440006, −3.74473073658080778760957102130, −3.63025883847804552651143556315, −2.26420958233220495466910118960, −2.02207295165448252042237538117, −1.60939559083074425450162986737, −1.31390476835185187048159087643,
1.31390476835185187048159087643, 1.60939559083074425450162986737, 2.02207295165448252042237538117, 2.26420958233220495466910118960, 3.63025883847804552651143556315, 3.74473073658080778760957102130, 3.80225388967424673482272440006, 3.80404894738793192763103248237, 4.78086893111247924514674194249, 5.09173433140503458662083098237, 5.33294689173421237124892140232, 5.34225973896219813567878461256, 5.82089910018840341214098844600, 6.09879207042107776860036845366, 6.50995405155104581071061562410, 6.76541269617338065495478811558, 7.24372312197739519344453805176, 7.58431261619051506237199756712, 7.73065782167575265624822082327, 7.76699232441108185938878281759, 8.436248783774728651012950058313, 8.538963608666549612557877313126, 8.630112055510370019449649098708, 8.740425684536063449337412965126, 9.322988967998302935069205618263
