| L(s) = 1 | + 2·5-s + 3·9-s − 4·11-s − 4·13-s − 6·17-s − 8·19-s + 5·25-s + 12·29-s − 8·31-s + 2·37-s − 4·41-s + 8·43-s + 6·45-s + 8·47-s − 6·53-s − 8·55-s − 6·61-s − 8·65-s − 4·67-s + 16·71-s + 10·73-s + 16·79-s + 16·83-s − 12·85-s − 6·89-s − 16·95-s + 12·97-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 9-s − 1.20·11-s − 1.10·13-s − 1.45·17-s − 1.83·19-s + 25-s + 2.22·29-s − 1.43·31-s + 0.328·37-s − 0.624·41-s + 1.21·43-s + 0.894·45-s + 1.16·47-s − 0.824·53-s − 1.07·55-s − 0.768·61-s − 0.992·65-s − 0.488·67-s + 1.89·71-s + 1.17·73-s + 1.80·79-s + 1.75·83-s − 1.30·85-s − 0.635·89-s − 1.64·95-s + 1.21·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.748193723\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.748193723\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 16 T + 177 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
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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
−10.75119470221633556636948128829, −9.924390313348500875756118113815, −9.888895426491069983272782602191, −9.206956048839568331853901495115, −8.877228451552243776462101443328, −8.480634415049145512604342918048, −7.87452900486457864826104507420, −7.51204657321655526559470134300, −6.98325129076072899329028696731, −6.42724374279470323287100445823, −6.40896181237153161910598432055, −5.61257489309429871597531707200, −4.99955155236562778776697661570, −4.66172332010624618152436203994, −4.39536815587148840809396507316, −3.57036606451058228395370250296, −2.71034590778072371638674923356, −2.19851134550428847796667157282, −1.99529843829270034797447060986, −0.64074531005130499193893225227,
0.64074531005130499193893225227, 1.99529843829270034797447060986, 2.19851134550428847796667157282, 2.71034590778072371638674923356, 3.57036606451058228395370250296, 4.39536815587148840809396507316, 4.66172332010624618152436203994, 4.99955155236562778776697661570, 5.61257489309429871597531707200, 6.40896181237153161910598432055, 6.42724374279470323287100445823, 6.98325129076072899329028696731, 7.51204657321655526559470134300, 7.87452900486457864826104507420, 8.480634415049145512604342918048, 8.877228451552243776462101443328, 9.206956048839568331853901495115, 9.888895426491069983272782602191, 9.924390313348500875756118113815, 10.75119470221633556636948128829
