| L(s) = 1 | + 4-s − 2·13-s + 4·17-s + 2·41-s − 2·49-s − 2·52-s − 2·53-s − 64-s + 4·68-s + 81-s + 2·89-s − 6·101-s + 4·109-s + 2·113-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 2·164-s + 167-s + 169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 4-s − 2·13-s + 4·17-s + 2·41-s − 2·49-s − 2·52-s − 2·53-s − 64-s + 4·68-s + 81-s + 2·89-s − 6·101-s + 4·109-s + 2·113-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 2·164-s + 167-s + 169-s + 173-s + 179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.368048730\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.368048730\) |
| \(L(1)\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| good | 3 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 11 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 17 | $C_1$$\times$$C_2^2$ | \( ( 1 - T )^{4}( 1 - T^{2} + T^{4} ) \) |
| 19 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 23 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 29 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 31 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T^{2} )^{2} \) |
| 43 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 53 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 59 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 61 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 67 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 71 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 89 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 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
−7.22081254810914189867636054548, −6.85037117926890792732293172329, −6.64746451159091968708224716896, −6.57761652874517360046492697326, −6.21192016445474981155147130363, −5.88943857932174513970587548715, −5.85786056802571380742879911382, −5.64550281169449422780692743525, −5.34117878271311864668705487997, −5.16117285225721120083328261762, −4.76724559401192212328546064644, −4.75403078698540938666394195366, −4.57769559182316272864547670566, −4.18042981337724032579443219280, −3.64751789296571546285687475697, −3.50854614965894052201297577187, −3.46823896389286503488581120930, −3.11850511041157738676478636571, −2.66059014142868064348918333139, −2.59158951538922230385405499127, −2.47009229086737084640932310274, −1.88299306277639599655467033469, −1.44436378728167524905838888226, −1.43104329102777798514480041287, −0.794107327105217107681433217234,
0.794107327105217107681433217234, 1.43104329102777798514480041287, 1.44436378728167524905838888226, 1.88299306277639599655467033469, 2.47009229086737084640932310274, 2.59158951538922230385405499127, 2.66059014142868064348918333139, 3.11850511041157738676478636571, 3.46823896389286503488581120930, 3.50854614965894052201297577187, 3.64751789296571546285687475697, 4.18042981337724032579443219280, 4.57769559182316272864547670566, 4.75403078698540938666394195366, 4.76724559401192212328546064644, 5.16117285225721120083328261762, 5.34117878271311864668705487997, 5.64550281169449422780692743525, 5.85786056802571380742879911382, 5.88943857932174513970587548715, 6.21192016445474981155147130363, 6.57761652874517360046492697326, 6.64746451159091968708224716896, 6.85037117926890792732293172329, 7.22081254810914189867636054548
