| L(s) = 1 | + 9-s − 4·13-s − 4·17-s + 25-s − 4·29-s + 14·37-s + 7·41-s + 5·49-s − 14·53-s + 8·73-s − 8·81-s − 7·89-s − 16·97-s − 17·101-s + 4·109-s − 14·113-s − 4·117-s + 5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 4·153-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 1/3·9-s − 1.10·13-s − 0.970·17-s + 1/5·25-s − 0.742·29-s + 2.30·37-s + 1.09·41-s + 5/7·49-s − 1.92·53-s + 0.936·73-s − 8/9·81-s − 0.741·89-s − 1.62·97-s − 1.69·101-s + 0.383·109-s − 1.31·113-s − 0.369·117-s + 5/11·121-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 − 0.323·153-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 924800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 924800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 17 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 45 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 99 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 127 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 95 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 89 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 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
−7.924593177907785500372677988148, −7.52866822488644854254761580537, −7.06373863794213516956924276805, −6.70903495018879873125423571838, −6.12569219894906451470456209879, −5.80199239104966680798964356788, −5.11625466554608399276132302634, −4.76568599817200906991156370475, −4.12020160041934980204199435019, −4.00268805362714537076296008824, −2.92392306174499579307357179791, −2.63490172155744030430619103615, −1.98888418769162094945280934062, −1.13382755993602998072466405852, 0,
1.13382755993602998072466405852, 1.98888418769162094945280934062, 2.63490172155744030430619103615, 2.92392306174499579307357179791, 4.00268805362714537076296008824, 4.12020160041934980204199435019, 4.76568599817200906991156370475, 5.11625466554608399276132302634, 5.80199239104966680798964356788, 6.12569219894906451470456209879, 6.70903495018879873125423571838, 7.06373863794213516956924276805, 7.52866822488644854254761580537, 7.924593177907785500372677988148
