| L(s) = 1 | − 3·2-s + 5·4-s + 8·7-s − 6·8-s + 9-s − 6·11-s − 24·14-s + 4·16-s + 6·17-s − 3·18-s + 6·19-s + 18·22-s + 12·23-s + 5·25-s + 40·28-s + 10·31-s − 18·34-s + 5·36-s − 4·37-s − 18·38-s − 32·43-s − 30·44-s − 36·46-s − 6·47-s + 34·49-s − 15·50-s + 6·53-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 5/2·4-s + 3.02·7-s − 2.12·8-s + 1/3·9-s − 1.80·11-s − 6.41·14-s + 16-s + 1.45·17-s − 0.707·18-s + 1.37·19-s + 3.83·22-s + 2.50·23-s + 25-s + 7.55·28-s + 1.79·31-s − 3.08·34-s + 5/6·36-s − 0.657·37-s − 2.91·38-s − 4.87·43-s − 4.52·44-s − 5.30·46-s − 0.875·47-s + 34/7·49-s − 2.12·50-s + 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.812292977\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.812292977\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} ) \) |
| 3 | $C_2^3$ | \( 1 - T^{2} - 8 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | $C_2$$\times$$C_2^2$ | \( ( 1 - p T^{2} )^{2}( 1 + p T^{2} + p^{2} T^{4} ) \) |
| 11 | $C_2^2$ | \( ( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 6 T + 13 T^{2} + 66 T^{3} - 372 T^{4} + 66 p T^{5} + 13 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 3 T - 10 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 12 T + 67 T^{2} - 372 T^{3} + 2088 T^{4} - 372 p T^{5} + 67 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 4 T - 17 T^{2} - 164 T^{3} - 872 T^{4} - 164 p T^{5} - 17 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 47 | $D_4\times C_2$ | \( 1 + 6 T - p T^{2} - 66 T^{3} + 2988 T^{4} - 66 p T^{5} - p^{3} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 6 T - 59 T^{2} + 66 T^{3} + 4308 T^{4} + 66 p T^{5} - 59 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 6 T - 71 T^{2} + 66 T^{3} + 5844 T^{4} + 66 p T^{5} - 71 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 3 T - 52 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 3 T - 58 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 8 T - 53 T^{2} + 232 T^{3} + 3688 T^{4} + 232 p T^{5} - 53 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 8 T - 65 T^{2} - 232 T^{3} + 5344 T^{4} - 232 p T^{5} - 65 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 89 | $C_2^3$ | \( 1 - 173 T^{2} + 22008 T^{4} - 173 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.20671776803264796450101768412, −6.79435386943882934250955087155, −6.75286636278374088542904287927, −6.64868455019575157540133013813, −6.12357493532839406634008462336, −5.79503195350725474188478093557, −5.71255287163473565103921145306, −5.22700521423171777629234952259, −5.03946911914954901355657644264, −4.91784962875383524269049550069, −4.90077805720536869481317503158, −4.67872993610053056427934158397, −4.65280073881114745836965557753, −3.66496200295194288997885792236, −3.61390989824593194726994036616, −3.46850859200000033925920931681, −2.79488112877270414127612213735, −2.78711187513178353260298122715, −2.70022235914273561840142761034, −1.96460054055032530093593554931, −1.78174561676838301276531874606, −1.52803282246173312726481778507, −1.17834069280827079155607748349, −0.988314660888264588742939175370, −0.46928506186351079857578087747,
0.46928506186351079857578087747, 0.988314660888264588742939175370, 1.17834069280827079155607748349, 1.52803282246173312726481778507, 1.78174561676838301276531874606, 1.96460054055032530093593554931, 2.70022235914273561840142761034, 2.78711187513178353260298122715, 2.79488112877270414127612213735, 3.46850859200000033925920931681, 3.61390989824593194726994036616, 3.66496200295194288997885792236, 4.65280073881114745836965557753, 4.67872993610053056427934158397, 4.90077805720536869481317503158, 4.91784962875383524269049550069, 5.03946911914954901355657644264, 5.22700521423171777629234952259, 5.71255287163473565103921145306, 5.79503195350725474188478093557, 6.12357493532839406634008462336, 6.64868455019575157540133013813, 6.75286636278374088542904287927, 6.79435386943882934250955087155, 7.20671776803264796450101768412
