L(s) = 1 | − 2·7-s + 9-s − 2·17-s − 10·23-s − 6·25-s − 12·31-s + 8·41-s + 8·47-s − 7·49-s − 2·63-s − 4·71-s + 6·73-s − 12·79-s − 8·81-s − 12·89-s + 20·97-s + 4·113-s + 4·119-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 2·153-s + 157-s + ⋯ |
L(s) = 1 | − 0.755·7-s + 1/3·9-s − 0.485·17-s − 2.08·23-s − 6/5·25-s − 2.15·31-s + 1.24·41-s + 1.16·47-s − 49-s − 0.251·63-s − 0.474·71-s + 0.702·73-s − 1.35·79-s − 8/9·81-s − 1.27·89-s + 2.03·97-s + 0.376·113-s + 0.366·119-s + 6/11·121-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 − 0.161·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + T^{2} \) |
good | 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 63 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
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
−9.336126258147034814112503800632, −9.064491317633067178100870937304, −8.389030956186099906416038465423, −7.79042982851671684778978336755, −7.42901332013629996889715545116, −6.90558712113642696385471185618, −6.13618729618275086988924178004, −5.92431601171042959229950212980, −5.32611062091362734112131005818, −4.34996062031826374295070956274, −3.99294556253432366313577205656, −3.39849180784135208243658224296, −2.41089010965825369147834256248, −1.74314978812294076481086490013, 0,
1.74314978812294076481086490013, 2.41089010965825369147834256248, 3.39849180784135208243658224296, 3.99294556253432366313577205656, 4.34996062031826374295070956274, 5.32611062091362734112131005818, 5.92431601171042959229950212980, 6.13618729618275086988924178004, 6.90558712113642696385471185618, 7.42901332013629996889715545116, 7.79042982851671684778978336755, 8.389030956186099906416038465423, 9.064491317633067178100870937304, 9.336126258147034814112503800632