| L(s) = 1 | − 4·2-s + 10·4-s − 4·7-s − 20·8-s − 2·9-s − 6·13-s + 16·14-s + 35·16-s + 8·18-s + 24·26-s − 40·28-s − 8·29-s − 56·32-s − 20·36-s − 4·37-s + 8·47-s + 16·49-s − 60·52-s + 80·56-s + 32·58-s + 40·61-s + 8·63-s + 84·64-s + 36·67-s + 40·72-s + 36·73-s + 16·74-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 5·4-s − 1.51·7-s − 7.07·8-s − 2/3·9-s − 1.66·13-s + 4.27·14-s + 35/4·16-s + 1.88·18-s + 4.70·26-s − 7.55·28-s − 1.48·29-s − 9.89·32-s − 3.33·36-s − 0.657·37-s + 1.16·47-s + 16/7·49-s − 8.32·52-s + 10.6·56-s + 4.20·58-s + 5.12·61-s + 1.00·63-s + 21/2·64-s + 4.39·67-s + 4.71·72-s + 4.21·73-s + 1.85·74-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5750509248\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5750509248\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{4} \) |
| 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| good | 7 | $D_{4}$ | \( ( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 190 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 56 T^{2} + 1774 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 31 | $D_4\times C_2$ | \( 1 - 88 T^{2} + 3790 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 2 T + 58 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 80 T^{2} + 3262 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 28 T^{2} + 2806 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 + 96 T^{2} + 7310 T^{4} + 96 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 184 T^{2} + 14814 T^{4} - 184 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 67 | $D_{4}$ | \( ( 1 - 18 T + 198 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 13894 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 18 T + 210 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 83 | $D_{4}$ | \( ( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 320 T^{2} + 41374 T^{4} - 320 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 18 T + 258 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.69910724033158905015173591054, −6.53178135869764567401411835335, −6.35405805250604899403397625377, −5.99888884514053454801064785707, −5.72622434175477454554832794838, −5.54730436924419985375046819052, −5.32076488195697486343266576098, −5.26028132892210687954445313447, −5.18417027208865692373149661992, −4.63463139422776371304037847504, −4.06589803563988310379639525664, −3.99767581267546126637115152324, −3.76299799816882694576718587296, −3.69960303142110675603502089834, −3.27905812853520514898777179710, −2.86503232580270665550045657231, −2.72749167829399010920066429819, −2.46502565400019801682730491767, −2.26272141564055991122881952317, −2.24892274606186142265257280938, −1.56846233603366561074380627698, −1.54247966423304117161995704142, −0.57908695306657818107987516020, −0.55184023070472585412947222210, −0.54591805223164192857838228813,
0.54591805223164192857838228813, 0.55184023070472585412947222210, 0.57908695306657818107987516020, 1.54247966423304117161995704142, 1.56846233603366561074380627698, 2.24892274606186142265257280938, 2.26272141564055991122881952317, 2.46502565400019801682730491767, 2.72749167829399010920066429819, 2.86503232580270665550045657231, 3.27905812853520514898777179710, 3.69960303142110675603502089834, 3.76299799816882694576718587296, 3.99767581267546126637115152324, 4.06589803563988310379639525664, 4.63463139422776371304037847504, 5.18417027208865692373149661992, 5.26028132892210687954445313447, 5.32076488195697486343266576098, 5.54730436924419985375046819052, 5.72622434175477454554832794838, 5.99888884514053454801064785707, 6.35405805250604899403397625377, 6.53178135869764567401411835335, 6.69910724033158905015173591054
