L(s) = 1 | + 4-s + 7-s − 13-s + 16-s − 2·19-s − 25-s + 28-s + 2·31-s − 2·43-s + 49-s − 52-s + 64-s + 2·73-s − 2·76-s − 2·79-s − 91-s − 2·97-s − 100-s + 112-s + ⋯ |
L(s) = 1 | + 4-s + 7-s − 13-s + 16-s − 2·19-s − 25-s + 28-s + 2·31-s − 2·43-s + 49-s − 52-s + 64-s + 2·73-s − 2·76-s − 2·79-s − 91-s − 2·97-s − 100-s + 112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.251383620\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.251383620\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 2 | \( ( 1 - T )( 1 + T ) \) |
| 5 | \( 1 + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 + T )^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( ( 1 - T )^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( ( 1 + T )^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )^{2} \) |
| 79 | \( ( 1 + T )^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( ( 1 + T )^{2} \) |
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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
−10.46831915002571710856588775373, −9.857099150156110565114598839409, −8.440098974431187269564506660352, −7.976986090834095275542773452756, −6.95215723222631208877289346109, −6.22411225716002505560485579821, −5.09912208755369812623947972946, −4.15646283745789568143023586820, −2.64628462579650265737685975477, −1.78858881088974432925030621222,
1.78858881088974432925030621222, 2.64628462579650265737685975477, 4.15646283745789568143023586820, 5.09912208755369812623947972946, 6.22411225716002505560485579821, 6.95215723222631208877289346109, 7.976986090834095275542773452756, 8.440098974431187269564506660352, 9.857099150156110565114598839409, 10.46831915002571710856588775373