| L(s) = 1 | + 2·5-s − 9-s − 12·11-s + 4·19-s − 25-s + 20·29-s − 4·31-s − 12·41-s − 2·45-s − 49-s − 24·55-s + 8·59-s − 4·61-s − 20·71-s − 16·79-s + 81-s − 12·89-s + 8·95-s + 12·99-s − 12·101-s + 28·109-s + 86·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1/3·9-s − 3.61·11-s + 0.917·19-s − 1/5·25-s + 3.71·29-s − 0.718·31-s − 1.87·41-s − 0.298·45-s − 1/7·49-s − 3.23·55-s + 1.04·59-s − 0.512·61-s − 2.37·71-s − 1.80·79-s + 1/9·81-s − 1.27·89-s + 0.820·95-s + 1.20·99-s − 1.19·101-s + 2.68·109-s + 7.81·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11289600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11289600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9690264030\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9690264030\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−9.068146146729537114272930467029, −8.453199198244193886678299260433, −8.100198088762637971247728741586, −7.78013129243434975082598804130, −7.32889798147739994935466139538, −7.00786101523006531554763313414, −6.61330454811213264582109507050, −5.92179092235838098545030041937, −5.77498393794199755421930674200, −5.44236370150368742515792824190, −5.01251073711345163935679115923, −4.73215144914997529827776690150, −4.45376067954748012345844059185, −3.45400992377609536384687406305, −3.07507896326722737879049119379, −2.64803013172638326691747812633, −2.58150475567522762056755070276, −1.86726545830743380500800015427, −1.20126884126363522542248350808, −0.29949943485577545580429800866,
0.29949943485577545580429800866, 1.20126884126363522542248350808, 1.86726545830743380500800015427, 2.58150475567522762056755070276, 2.64803013172638326691747812633, 3.07507896326722737879049119379, 3.45400992377609536384687406305, 4.45376067954748012345844059185, 4.73215144914997529827776690150, 5.01251073711345163935679115923, 5.44236370150368742515792824190, 5.77498393794199755421930674200, 5.92179092235838098545030041937, 6.61330454811213264582109507050, 7.00786101523006531554763313414, 7.32889798147739994935466139538, 7.78013129243434975082598804130, 8.100198088762637971247728741586, 8.453199198244193886678299260433, 9.068146146729537114272930467029
