| L(s) = 1 | − 2-s + 4-s − 8-s + 11-s + 16-s + 17-s − 19-s − 22-s + 25-s − 32-s − 34-s + 38-s + 41-s − 43-s + 44-s + 49-s − 50-s + 59-s + 64-s − 67-s + 68-s − 73-s − 76-s − 82-s − 2·83-s + 86-s − 88-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s − 8-s + 11-s + 16-s + 17-s − 19-s − 22-s + 25-s − 32-s − 34-s + 38-s + 41-s − 43-s + 44-s + 49-s − 50-s + 59-s + 64-s − 67-s + 68-s − 73-s − 76-s − 82-s − 2·83-s + 86-s − 88-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6532441654\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6532441654\) |
| \(L(1)\) |
|
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 | \( 1 \) |
| good | 5 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 - T + T^{2} \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( 1 - T + T^{2} \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( 1 - T + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 + T )^{2} \) |
| 89 | \( ( 1 + T )^{2} \) |
| 97 | \( 1 + T + T^{2} \) |
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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
−10.60341475152256943221806619334, −9.874347180030868415081602803923, −8.982486746031989707936807894926, −8.351644822249025280774420685553, −7.29671137085968494935281080661, −6.54439724480456005171908411935, −5.59468296879334546829024752782, −4.07812634788854752097901427150, −2.81441074757590216582168505230, −1.36724421983330013271266289894,
1.36724421983330013271266289894, 2.81441074757590216582168505230, 4.07812634788854752097901427150, 5.59468296879334546829024752782, 6.54439724480456005171908411935, 7.29671137085968494935281080661, 8.351644822249025280774420685553, 8.982486746031989707936807894926, 9.874347180030868415081602803923, 10.60341475152256943221806619334
