| L(s) = 1 | − 26·7-s + 9·9-s + 16·16-s + 74·19-s + 92·31-s − 52·37-s + 409·49-s − 234·63-s − 244·67-s + 286·73-s + 4·97-s + 314·103-s − 428·109-s − 416·112-s + 127-s + 131-s − 1.92e3·133-s + 137-s + 139-s + 144·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 191·169-s + 666·171-s + ⋯ |
| L(s) = 1 | − 3.71·7-s + 9-s + 16-s + 3.89·19-s + 2.96·31-s − 1.40·37-s + 8.34·49-s − 3.71·63-s − 3.64·67-s + 3.91·73-s + 4/97·97-s + 3.04·103-s − 3.92·109-s − 3.71·112-s + 0.00787·127-s + 0.00763·131-s − 14.4·133-s + 0.00729·137-s + 0.00719·139-s + 144-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 1.13·169-s + 3.89·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.095333348\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.095333348\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2^2$ | \( 1 - p^{2} T^{2} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 13 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 191 T^{2} + p^{4} T^{4} \) |
| good | 2 | $C_2^3$ | \( 1 - p^{4} T^{4} + p^{8} T^{8} \) |
| 5 | $C_2^3$ | \( 1 - p^{4} T^{4} + p^{8} T^{8} \) |
| 11 | $C_2^3$ | \( 1 - p^{4} T^{4} + p^{8} T^{8} \) |
| 17 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2$$\times$$C_2^2$ | \( ( 1 - 37 T + p^{2} T^{2} )^{2}( 1 - 46 T^{2} + p^{4} T^{4} ) \) |
| 23 | $C_2^2$ | \( ( 1 - p^{2} T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 31 | $C_2$$\times$$C_2^2$ | \( ( 1 - 46 T + p^{2} T^{2} )^{2}( 1 + 1559 T^{2} + p^{4} T^{4} ) \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 + 26 T + p^{2} T^{2} )^{2}( 1 - 529 T^{2} + p^{4} T^{4} ) \) |
| 41 | $C_2^2$ | \( ( 1 + p^{4} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 61 T + p^{2} T^{2} )^{2}( 1 + 61 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - p^{4} T^{4} + p^{8} T^{8} \) |
| 53 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2}( 1 + p T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - p^{4} T^{4} + p^{8} T^{8} \) |
| 61 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 5233 T^{2} + p^{4} T^{4} )( 1 + 7199 T^{2} + p^{4} T^{4} ) \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + 122 T + p^{2} T^{2} )^{2}( 1 + 2903 T^{2} + p^{4} T^{4} ) \) |
| 71 | $C_2^2$ | \( ( 1 + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$$\times$$C_2^2$ | \( ( 1 - 143 T + p^{2} T^{2} )^{2}( 1 - 8542 T^{2} + p^{4} T^{4} ) \) |
| 79 | $C_2^2$$\times$$C_2^2$ | \( ( 1 + 4679 T^{2} + p^{4} T^{4} )( 1 + 7682 T^{2} + p^{4} T^{4} ) \) |
| 83 | $C_2^2$ | \( ( 1 + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - p^{4} T^{4} + p^{8} T^{8} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{2}( 1 - 18814 T^{2} + p^{4} T^{4} ) \) |
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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
−8.456877367360934819728371302841, −8.057241247918567922865059087081, −7.80644143397199410598413548073, −7.52084028732876552545151684579, −7.50285302011265554200754471710, −7.02219538654136790419061600634, −6.67995220263102441020505460645, −6.61577687221184757126331457312, −6.58497047128473889645462619931, −6.05313154428967150117842637269, −5.72634666096984874584827920725, −5.59251918332251362482048771040, −5.28275838742204187062423785640, −4.97867573459702136462372681095, −4.33071442408030828455884493177, −4.29730332514442438383606357561, −3.70145996695869363837369311427, −3.27483414690245395347049674462, −3.21819834755159208209555043360, −3.19847851783495635127065076379, −2.70863256914225057017152067571, −2.20186699085188298556118703880, −1.16673781659977757868915320472, −1.06606180979957026972818545634, −0.45199573894034998163463890037,
0.45199573894034998163463890037, 1.06606180979957026972818545634, 1.16673781659977757868915320472, 2.20186699085188298556118703880, 2.70863256914225057017152067571, 3.19847851783495635127065076379, 3.21819834755159208209555043360, 3.27483414690245395347049674462, 3.70145996695869363837369311427, 4.29730332514442438383606357561, 4.33071442408030828455884493177, 4.97867573459702136462372681095, 5.28275838742204187062423785640, 5.59251918332251362482048771040, 5.72634666096984874584827920725, 6.05313154428967150117842637269, 6.58497047128473889645462619931, 6.61577687221184757126331457312, 6.67995220263102441020505460645, 7.02219538654136790419061600634, 7.50285302011265554200754471710, 7.52084028732876552545151684579, 7.80644143397199410598413548073, 8.057241247918567922865059087081, 8.456877367360934819728371302841
