| L(s) = 1 | − 2-s − 7-s + 8-s + 11-s + 14-s − 16-s − 22-s − 23-s + 29-s + 37-s + 43-s + 46-s + 49-s + 2·53-s − 56-s − 58-s + 64-s + 67-s + 71-s − 74-s − 77-s − 79-s − 86-s + 88-s − 98-s − 2·106-s + 2·107-s + ⋯ |
| L(s) = 1 | − 2-s − 7-s + 8-s + 11-s + 14-s − 16-s − 22-s − 23-s + 29-s + 37-s + 43-s + 46-s + 49-s + 2·53-s − 56-s − 58-s + 64-s + 67-s + 71-s − 74-s − 77-s − 79-s − 86-s + 88-s − 98-s − 2·106-s + 2·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5590472548\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5590472548\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 2 | \( 1 + T + T^{2} \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( 1 - T + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( 1 - T + T^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( 1 - T + T^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( 1 - T + T^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 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
−9.623381387884658241348518171947, −8.910971909796422676638143030997, −8.247685610421616020023026468256, −7.30748434746590630575149047472, −6.58671895139458492128757551006, −5.74531244597510999207877374089, −4.43304395166399631448140218903, −3.71821822809850346474350049207, −2.36241087978402030020583646858, −0.929678056933204654072478137645,
0.929678056933204654072478137645, 2.36241087978402030020583646858, 3.71821822809850346474350049207, 4.43304395166399631448140218903, 5.74531244597510999207877374089, 6.58671895139458492128757551006, 7.30748434746590630575149047472, 8.247685610421616020023026468256, 8.910971909796422676638143030997, 9.623381387884658241348518171947
