L(s) = 1 | − 2-s + 5-s + 8-s − 10-s − 16-s − 17-s − 19-s − 23-s + 25-s − 31-s + 34-s + 38-s + 40-s + 46-s + 2·47-s + 49-s − 50-s − 53-s − 61-s + 62-s + 64-s − 79-s − 80-s − 83-s − 85-s − 2·94-s − 95-s + ⋯ |
L(s) = 1 | − 2-s + 5-s + 8-s − 10-s − 16-s − 17-s − 19-s − 23-s + 25-s − 31-s + 34-s + 38-s + 40-s + 46-s + 2·47-s + 49-s − 50-s − 53-s − 61-s + 62-s + 64-s − 79-s − 80-s − 83-s − 85-s − 2·94-s − 95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4163687655\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4163687655\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
good | 2 | \( 1 + T + T^{2} \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 + T + T^{2} \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( 1 + T + T^{2} \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 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
−13.51579545588941529611973298262, −12.59449163938264305142168973101, −10.98652682515726003864537765386, −10.25584206721761290089387432829, −9.253566177983878475275416700587, −8.548308440781228384134559841616, −7.19935214621441948680617539145, −5.91615283308312356244034077336, −4.40323847289512931191199822637, −2.04028622872293924908533618653,
2.04028622872293924908533618653, 4.40323847289512931191199822637, 5.91615283308312356244034077336, 7.19935214621441948680617539145, 8.548308440781228384134559841616, 9.253566177983878475275416700587, 10.25584206721761290089387432829, 10.98652682515726003864537765386, 12.59449163938264305142168973101, 13.51579545588941529611973298262