| L(s) = 1 | − 5-s + 7-s + 9-s + 11-s − 17-s − 19-s − 2·23-s − 35-s + 43-s − 45-s + 47-s − 55-s − 61-s + 63-s − 73-s + 77-s + 81-s − 2·83-s + 85-s + 95-s + 99-s + 2·101-s + 2·115-s − 119-s + ⋯ |
| L(s) = 1 | − 5-s + 7-s + 9-s + 11-s − 17-s − 19-s − 2·23-s − 35-s + 43-s − 45-s + 47-s − 55-s − 61-s + 63-s − 73-s + 77-s + 81-s − 2·83-s + 85-s + 95-s + 99-s + 2·101-s + 2·115-s − 119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7687285364\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7687285364\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( ( 1 - T )( 1 + T ) \) |
| 5 | \( 1 + T + T^{2} \) |
| 7 | \( 1 - T + T^{2} \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 + T + T^{2} \) |
| 23 | \( ( 1 + T )^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( 1 - T + T^{2} \) |
| 47 | \( 1 - T + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 + T )^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.87623690395613953083244472332, −11.16677617376553065237654652648, −10.20644809692228960587710472989, −8.975755844324787350507900962127, −8.078429734036841411010063532632, −7.26473463181749542271796798285, −6.14914763837325908462523917368, −4.39370561154733445874325165816, −4.07349989989523942983386942659, −1.86990732021995692418250624709,
1.86990732021995692418250624709, 4.07349989989523942983386942659, 4.39370561154733445874325165816, 6.14914763837325908462523917368, 7.26473463181749542271796798285, 8.078429734036841411010063532632, 8.975755844324787350507900962127, 10.20644809692228960587710472989, 11.16677617376553065237654652648, 11.87623690395613953083244472332
