| L(s) = 1 | + 2·3-s − 4·5-s + 6·7-s + 2·9-s + 4·11-s − 6·13-s − 8·15-s + 2·17-s + 12·21-s + 2·23-s + 11·25-s + 6·27-s + 8·33-s − 24·35-s + 2·37-s − 12·39-s + 20·41-s − 10·43-s − 8·45-s − 6·47-s + 18·49-s + 4·51-s − 10·53-s − 16·55-s + 12·63-s + 24·65-s + 2·67-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.78·5-s + 2.26·7-s + 2/3·9-s + 1.20·11-s − 1.66·13-s − 2.06·15-s + 0.485·17-s + 2.61·21-s + 0.417·23-s + 11/5·25-s + 1.15·27-s + 1.39·33-s − 4.05·35-s + 0.328·37-s − 1.92·39-s + 3.12·41-s − 1.52·43-s − 1.19·45-s − 0.875·47-s + 18/7·49-s + 0.560·51-s − 1.37·53-s − 2.15·55-s + 1.51·63-s + 2.97·65-s + 0.244·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.469760858\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.469760858\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 138 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−9.517903836074491164888341976728, −9.464062785573687454981868709775, −8.961028642058998952491036516737, −8.434780925515295408288556011918, −8.286360693471048380898984215120, −7.69993112328264032091837544466, −7.63928529242605102454852811328, −7.47452493913816062160691201612, −6.62253915419771897206932509035, −6.48760004088276344169223511844, −5.37051519172201753730823556759, −5.00160488029321986850256738771, −4.62815617944922678664482831841, −4.32206740114971620348748626768, −3.87790605644655920958436775839, −3.26897770244475509297242269435, −2.78168196671581252471151035123, −2.16445420791154516633389851178, −1.46952488541854550988251911405, −0.797254719091624168491719672082,
0.797254719091624168491719672082, 1.46952488541854550988251911405, 2.16445420791154516633389851178, 2.78168196671581252471151035123, 3.26897770244475509297242269435, 3.87790605644655920958436775839, 4.32206740114971620348748626768, 4.62815617944922678664482831841, 5.00160488029321986850256738771, 5.37051519172201753730823556759, 6.48760004088276344169223511844, 6.62253915419771897206932509035, 7.47452493913816062160691201612, 7.63928529242605102454852811328, 7.69993112328264032091837544466, 8.286360693471048380898984215120, 8.434780925515295408288556011918, 8.961028642058998952491036516737, 9.464062785573687454981868709775, 9.517903836074491164888341976728
