L(s) = 1 | + 2·9-s − 10·11-s − 14·19-s − 10·29-s + 4·31-s + 22·41-s + 13·49-s + 28·59-s + 20·61-s − 20·71-s − 14·79-s − 5·81-s − 20·89-s − 20·99-s + 36·101-s − 4·109-s + 53·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 25·169-s + ⋯ |
L(s) = 1 | + 2/3·9-s − 3.01·11-s − 3.21·19-s − 1.85·29-s + 0.718·31-s + 3.43·41-s + 13/7·49-s + 3.64·59-s + 2.56·61-s − 2.37·71-s − 1.57·79-s − 5/9·81-s − 2.11·89-s − 2.01·99-s + 3.58·101-s − 0.383·109-s + 4.81·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.92·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21160000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9709599998\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9709599998\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 23 | $C_2$ | \( 1 + T^{2} \) |
good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 97 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 59 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 178 T^{2} + p^{2} T^{4} \) |
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
−8.424787667637716630155467492977, −8.264269131570749903758974397380, −7.59778837321851120194824909342, −7.55609953443471149342391748458, −7.09066014002348235488003929050, −6.89796241843628698252813327936, −6.20050033256577282849210668107, −5.73041631065807757287145543299, −5.70166997028613125475032034557, −5.34843722596875682120949105119, −4.57437683728827654822942559735, −4.51207667870896595434632516649, −3.92718986990685228776643370656, −3.86423869656204061429269847821, −2.83057015664671695292213197785, −2.58279445913772675595450311801, −2.16377438635815451101202007456, −2.07690155642178728129418253807, −0.927615109624252587194619249141, −0.31162210229232758839415089723,
0.31162210229232758839415089723, 0.927615109624252587194619249141, 2.07690155642178728129418253807, 2.16377438635815451101202007456, 2.58279445913772675595450311801, 2.83057015664671695292213197785, 3.86423869656204061429269847821, 3.92718986990685228776643370656, 4.51207667870896595434632516649, 4.57437683728827654822942559735, 5.34843722596875682120949105119, 5.70166997028613125475032034557, 5.73041631065807757287145543299, 6.20050033256577282849210668107, 6.89796241843628698252813327936, 7.09066014002348235488003929050, 7.55609953443471149342391748458, 7.59778837321851120194824909342, 8.264269131570749903758974397380, 8.424787667637716630155467492977