| L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 4·9-s + 10-s − 3·13-s + 16-s − 6·17-s − 4·18-s − 20-s − 4·25-s + 3·26-s − 6·29-s − 32-s + 6·34-s + 4·36-s + 4·37-s + 40-s − 4·45-s + 14·49-s + 4·50-s − 3·52-s + 6·58-s − 2·61-s + 64-s + 3·65-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 4/3·9-s + 0.316·10-s − 0.832·13-s + 1/4·16-s − 1.45·17-s − 0.942·18-s − 0.223·20-s − 4/5·25-s + 0.588·26-s − 1.11·29-s − 0.176·32-s + 1.02·34-s + 2/3·36-s + 0.657·37-s + 0.158·40-s − 0.596·45-s + 2·49-s + 0.565·50-s − 0.416·52-s + 0.787·58-s − 0.256·61-s + 1/8·64-s + 0.372·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2080 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2080 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4803631948\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4803631948\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 4 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 76 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 80 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−13.20859068784444078804919596956, −12.59086967665366017226981995880, −12.01019791255937568083245043080, −11.34656975509688312402616161032, −10.74781547782242050869487639672, −10.08922399836353334407924537776, −9.480918778573358190832723629444, −8.933390642891826377839652184385, −8.004172495143742733490166513119, −7.34239524010294972057117625494, −6.93300330028434354687859432943, −5.91298812586149253328721774965, −4.68473046101937618803253601661, −3.89295282825188061140108988893, −2.19543188028031805089419316236,
2.19543188028031805089419316236, 3.89295282825188061140108988893, 4.68473046101937618803253601661, 5.91298812586149253328721774965, 6.93300330028434354687859432943, 7.34239524010294972057117625494, 8.004172495143742733490166513119, 8.933390642891826377839652184385, 9.480918778573358190832723629444, 10.08922399836353334407924537776, 10.74781547782242050869487639672, 11.34656975509688312402616161032, 12.01019791255937568083245043080, 12.59086967665366017226981995880, 13.20859068784444078804919596956
