| L(s) = 1 | + 26·3-s − 250·5-s + 686·7-s + 729·9-s − 2.52e3·11-s − 2.77e3·13-s − 6.50e3·15-s + 754·17-s + 1.78e4·21-s + 4.68e4·25-s + 3.92e4·27-s + 4.58e4·29-s − 6.55e4·33-s − 1.71e5·35-s − 7.21e4·39-s − 1.82e5·45-s − 1.75e5·47-s + 3.52e5·49-s + 1.96e4·51-s + 6.30e5·55-s + 5.00e5·63-s + 6.93e5·65-s − 6.04e4·71-s + 1.00e6·73-s + 1.21e6·75-s − 1.73e6·77-s + 9.30e5·79-s + ⋯ |
| L(s) = 1 | + 0.962·3-s − 2·5-s + 2·7-s + 9-s − 1.89·11-s − 1.26·13-s − 1.92·15-s + 0.153·17-s + 1.92·21-s + 3·25-s + 1.99·27-s + 1.88·29-s − 1.82·33-s − 4·35-s − 1.21·39-s − 2·45-s − 1.69·47-s + 3·49-s + 0.147·51-s + 3.78·55-s + 2·63-s + 2.52·65-s − 0.168·71-s + 2.59·73-s + 26/9·75-s − 3.78·77-s + 1.88·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.485508267\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.485508267\) |
| \(L(4)\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p^{3} T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 26 T - 53 T^{2} - 26 p^{6} T^{3} + p^{12} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2522 T + 4588923 T^{2} + 2522 p^{6} T^{3} + p^{12} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 2774 T + 2868267 T^{2} + 2774 p^{6} T^{3} + p^{12} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 754 T - 23569053 T^{2} - 754 p^{6} T^{3} + p^{12} T^{4} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 45862 T + 1508499723 T^{2} - 45862 p^{6} T^{3} + p^{12} T^{4} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 175646 T + 20072301987 T^{2} + 175646 p^{6} T^{3} + p^{12} T^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 30238 T + p^{6} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 504254 T + p^{6} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 930382 T + 622523210403 T^{2} - 930382 p^{6} T^{3} + p^{12} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 1141306 T + p^{6} T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{3} T )^{2}( 1 + p^{3} T )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 897874 T - 26794285053 T^{2} - 897874 p^{6} T^{3} + p^{12} T^{4} \) |
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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
−12.27250974023290203443216568510, −11.86731993842168870176901677080, −11.20816362471207635724532811854, −10.82021548314364337943440565113, −10.33091597503014912140853445577, −9.828716597512453226243906053020, −8.728649285330973306848460153327, −8.374232724014088140722784519032, −8.087850895508946410660609353779, −7.67449710350352034955685380600, −7.32265268409782504924753755544, −6.67652214332652960330933087816, −5.05263784043602835103693506163, −4.94728098407777819146661896135, −4.54323037129017165436833730927, −3.71498564895203334839690219058, −2.77626417141834891120328056919, −2.48519402973338931580020224571, −1.30390093134689117611902786935, −0.48615148914879313987868127656,
0.48615148914879313987868127656, 1.30390093134689117611902786935, 2.48519402973338931580020224571, 2.77626417141834891120328056919, 3.71498564895203334839690219058, 4.54323037129017165436833730927, 4.94728098407777819146661896135, 5.05263784043602835103693506163, 6.67652214332652960330933087816, 7.32265268409782504924753755544, 7.67449710350352034955685380600, 8.087850895508946410660609353779, 8.374232724014088140722784519032, 8.728649285330973306848460153327, 9.828716597512453226243906053020, 10.33091597503014912140853445577, 10.82021548314364337943440565113, 11.20816362471207635724532811854, 11.86731993842168870176901677080, 12.27250974023290203443216568510
