| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 7-s + 9-s + 2·11-s + 12-s + 14-s − 18-s − 19-s − 21-s − 2·22-s + 23-s + 4·25-s − 28-s + 32-s + 2·33-s + 36-s + 38-s − 41-s + 42-s + 2·44-s − 46-s + 49-s − 4·50-s − 2·53-s + ⋯ |
| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 7-s + 9-s + 2·11-s + 12-s + 14-s − 18-s − 19-s − 21-s − 2·22-s + 23-s + 4·25-s − 28-s + 32-s + 2·33-s + 36-s + 38-s − 41-s + 42-s + 2·44-s − 46-s + 49-s − 4·50-s − 2·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.693515218\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.693515218\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 13 | | \( 1 \) |
| good | 2 | $C_4\times C_2$ | \( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} \) |
| 3 | $C_4\times C_2$ | \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 7 | $C_4\times C_2$ | \( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 19 | $C_4\times C_2$ | \( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} \) |
| 23 | $C_4\times C_2$ | \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 41 | $C_4\times C_2$ | \( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 53 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 73 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 83 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
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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
−6.91789388936982228852827860508, −6.53887047486827115583102688257, −6.38495362338698665169902262518, −6.36562870835256555025457009635, −6.29443076873478273602733820603, −5.90687373248853390396662274767, −5.56225393262291710788002689167, −5.01803144777260685952954823991, −4.96060050314026756305114350412, −4.87338470015644872633230847770, −4.78816060941540580756413394176, −4.25849949411059740883304861699, −4.16613542139994979735005208055, −3.63255972702343510523761362683, −3.58571279108148243151393541180, −3.52785419925560541736864567178, −3.26144336822844253568526095690, −2.79834940735570818638491562373, −2.56247211382127301731088944395, −2.42655582557872478330818560558, −2.19796428821317423964597392643, −1.56014463873826563548800012018, −1.35514688192140236055118122844, −1.20626941916860746336992408420, −0.897805846323087410129088445687,
0.897805846323087410129088445687, 1.20626941916860746336992408420, 1.35514688192140236055118122844, 1.56014463873826563548800012018, 2.19796428821317423964597392643, 2.42655582557872478330818560558, 2.56247211382127301731088944395, 2.79834940735570818638491562373, 3.26144336822844253568526095690, 3.52785419925560541736864567178, 3.58571279108148243151393541180, 3.63255972702343510523761362683, 4.16613542139994979735005208055, 4.25849949411059740883304861699, 4.78816060941540580756413394176, 4.87338470015644872633230847770, 4.96060050314026756305114350412, 5.01803144777260685952954823991, 5.56225393262291710788002689167, 5.90687373248853390396662274767, 6.29443076873478273602733820603, 6.36562870835256555025457009635, 6.38495362338698665169902262518, 6.53887047486827115583102688257, 6.91789388936982228852827860508
