| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s − 12-s − 2·13-s + 16-s + 18-s − 23-s − 24-s − 25-s − 2·26-s − 27-s + 32-s + 36-s + 2·39-s − 46-s + 2·47-s − 48-s − 49-s − 50-s − 2·52-s − 54-s + 2·59-s + 64-s + ⋯ |
| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s − 12-s − 2·13-s + 16-s + 18-s − 23-s − 24-s − 25-s − 2·26-s − 27-s + 32-s + 36-s + 2·39-s − 46-s + 2·47-s − 48-s − 49-s − 50-s − 2·52-s − 54-s + 2·59-s + 64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9935342802\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9935342802\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 + T )^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( ( 1 - T )^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( ( 1 - T )^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( ( 1 + T )^{2} \) |
| 73 | \( ( 1 - T )^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.10785865755557287761980673259, −11.55009886897249101253129072936, −10.39256687254628662397292127470, −9.729189916338952454796774846704, −7.75485496510642676156134406135, −6.99692796576244713645341749392, −5.88862139374234201961255239624, −5.02813123257204501746520012419, −4.04599332383850795239322337996, −2.24544852622770945663253188417,
2.24544852622770945663253188417, 4.04599332383850795239322337996, 5.02813123257204501746520012419, 5.88862139374234201961255239624, 6.99692796576244713645341749392, 7.75485496510642676156134406135, 9.729189916338952454796774846704, 10.39256687254628662397292127470, 11.55009886897249101253129072936, 12.10785865755557287761980673259
