| L(s) = 1 | + 4-s − 9-s + 16-s − 17-s − 19-s + 25-s − 36-s − 2·43-s − 2·47-s + 49-s + 64-s − 68-s − 76-s + 81-s + 2·83-s + 100-s + 2·101-s + ⋯ |
| L(s) = 1 | + 4-s − 9-s + 16-s − 17-s − 19-s + 25-s − 36-s − 2·43-s − 2·47-s + 49-s + 64-s − 68-s − 76-s + 81-s + 2·83-s + 100-s + 2·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8924921034\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8924921034\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( ( 1 - T )( 1 + T ) \) |
| 3 | \( 1 + T^{2} \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( ( 1 + T )^{2} \) |
| 47 | \( ( 1 + T )^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( ( 1 - T )^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( 1 + T^{2} \) |
| 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
−11.64966816165104568761491584925, −11.05048287482994918132775614845, −10.22672626968250785864872907358, −8.880549405771016938219218871216, −8.118659427712678214345444546269, −6.84327250477387608667083132062, −6.19151968678992415308160504190, −4.92570925268475506399702373053, −3.30136659857881441198667195029, −2.13564136646477048803413760845,
2.13564136646477048803413760845, 3.30136659857881441198667195029, 4.92570925268475506399702373053, 6.19151968678992415308160504190, 6.84327250477387608667083132062, 8.118659427712678214345444546269, 8.880549405771016938219218871216, 10.22672626968250785864872907358, 11.05048287482994918132775614845, 11.64966816165104568761491584925
