| L(s) = 1 | + 4·7-s + 2·9-s − 4·17-s + 12·23-s − 2·25-s − 4·41-s − 8·47-s + 2·49-s + 8·63-s + 4·71-s + 4·73-s + 24·79-s − 5·81-s + 4·89-s − 4·97-s − 12·103-s − 12·113-s − 16·119-s − 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 8·153-s + 157-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 2/3·9-s − 0.970·17-s + 2.50·23-s − 2/5·25-s − 0.624·41-s − 1.16·47-s + 2/7·49-s + 1.00·63-s + 0.474·71-s + 0.468·73-s + 2.70·79-s − 5/9·81-s + 0.423·89-s − 0.406·97-s − 1.18·103-s − 1.12·113-s − 1.46·119-s − 1.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 0.646·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.790018988\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.790018988\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.861546300263885717610995941498, −9.257445930308445842424916850697, −8.962509216952340219493373040213, −8.274502461895136090024958629613, −7.933588155442246056969085945507, −7.37927148774689412811120971737, −6.68005114138485920502163249539, −6.52870926368114517685804147148, −5.30363921268608459919205472857, −5.07940251636056314821122674916, −4.56241426928615040628939688911, −3.87473316277148274342156283239, −2.98550417606470505941463917702, −2.04007410150240677757019830272, −1.25622647594079178419903930160,
1.25622647594079178419903930160, 2.04007410150240677757019830272, 2.98550417606470505941463917702, 3.87473316277148274342156283239, 4.56241426928615040628939688911, 5.07940251636056314821122674916, 5.30363921268608459919205472857, 6.52870926368114517685804147148, 6.68005114138485920502163249539, 7.37927148774689412811120971737, 7.933588155442246056969085945507, 8.274502461895136090024958629613, 8.962509216952340219493373040213, 9.257445930308445842424916850697, 9.861546300263885717610995941498
