| L(s) = 1 | − 3-s + 3·7-s + 13-s − 3·21-s − 2·25-s + 27-s − 39-s − 43-s + 5·49-s − 61-s − 3·67-s + 2·75-s − 2·79-s − 81-s + 3·91-s − 3·97-s + 2·103-s + 121-s + 127-s + 129-s + 131-s + 137-s + 139-s − 5·147-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 3-s + 3·7-s + 13-s − 3·21-s − 2·25-s + 27-s − 39-s − 43-s + 5·49-s − 61-s − 3·67-s + 2·75-s − 2·79-s − 81-s + 3·91-s − 3·97-s + 2·103-s + 121-s + 127-s + 129-s + 131-s + 137-s + 139-s − 5·147-s + 149-s + 151-s + 157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 389376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 389376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8143423264\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8143423264\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 13 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.16326648644933549402327645131, −10.83262064549757261779025819985, −10.29586929035861436136625478662, −9.998511309693193538274555644499, −9.176867286509257525104333490791, −8.668807895706193204494562136858, −8.390855695567502536544133074677, −8.065784939512042983425540800756, −7.39264452533451562382393925326, −7.37301608915810004823649888907, −6.32871874525778602190014383538, −5.97583254475485776225850655691, −5.54484731255588674475720137007, −5.15284744389746398276964585931, −4.46788908484202529413591837651, −4.41813693880371658290269596008, −3.57909551619564285295912477576, −2.61655853986043007549936872830, −1.59513767482707627453932159978, −1.50515612135362487738805585777,
1.50515612135362487738805585777, 1.59513767482707627453932159978, 2.61655853986043007549936872830, 3.57909551619564285295912477576, 4.41813693880371658290269596008, 4.46788908484202529413591837651, 5.15284744389746398276964585931, 5.54484731255588674475720137007, 5.97583254475485776225850655691, 6.32871874525778602190014383538, 7.37301608915810004823649888907, 7.39264452533451562382393925326, 8.065784939512042983425540800756, 8.390855695567502536544133074677, 8.668807895706193204494562136858, 9.176867286509257525104333490791, 9.998511309693193538274555644499, 10.29586929035861436136625478662, 10.83262064549757261779025819985, 11.16326648644933549402327645131
