| L(s) = 1 | + 2·7-s + 2·11-s − 4·17-s + 2·23-s + 25-s − 2·31-s + 2·41-s + 2·49-s + 4·53-s + 2·67-s + 4·71-s + 4·73-s + 4·77-s − 2·101-s − 2·103-s − 4·107-s − 2·113-s − 8·119-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 4·161-s + ⋯ |
| L(s) = 1 | + 2·7-s + 2·11-s − 4·17-s + 2·23-s + 25-s − 2·31-s + 2·41-s + 2·49-s + 4·53-s + 2·67-s + 4·71-s + 4·73-s + 4·77-s − 2·101-s − 2·103-s − 4·107-s − 2·113-s − 8·119-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 4·161-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.875142269\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.875142269\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| good | 7 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 13 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 29 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 43 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 47 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 61 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 97 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
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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
−6.91757879496640183242624868892, −6.58839572943138311146875250768, −6.56114341550822524922673303922, −6.53419131199815486750866616118, −6.21749444399808995199126167357, −5.68634658767595129188056492002, −5.44038563481239592667121936743, −5.33612847460532026130606893651, −5.16245144681585178975938557281, −5.01201311290288263849517816199, −4.67501408808301547431104228098, −4.50670400782096265321526619677, −4.25095611198084798298916075074, −3.94237527514091366412939739054, −3.74351710674456036782814021658, −3.71159123079632548022904529547, −3.66200726305806221279222612118, −2.63982610821641598504701724611, −2.50649060940919929237898850424, −2.47011711810938042525988587645, −2.21253849749439331465534944653, −1.98630893470552478282824095994, −1.33007660637035564411500531200, −1.15008247787606334460081597165, −0.965515409643568353301506364935,
0.965515409643568353301506364935, 1.15008247787606334460081597165, 1.33007660637035564411500531200, 1.98630893470552478282824095994, 2.21253849749439331465534944653, 2.47011711810938042525988587645, 2.50649060940919929237898850424, 2.63982610821641598504701724611, 3.66200726305806221279222612118, 3.71159123079632548022904529547, 3.74351710674456036782814021658, 3.94237527514091366412939739054, 4.25095611198084798298916075074, 4.50670400782096265321526619677, 4.67501408808301547431104228098, 5.01201311290288263849517816199, 5.16245144681585178975938557281, 5.33612847460532026130606893651, 5.44038563481239592667121936743, 5.68634658767595129188056492002, 6.21749444399808995199126167357, 6.53419131199815486750866616118, 6.56114341550822524922673303922, 6.58839572943138311146875250768, 6.91757879496640183242624868892
