| L(s) = 1 | − 5.27·7-s − 6.31·13-s + 3.46·19-s − 5·25-s − 4.52·31-s − 11.9·37-s − 10.3·43-s + 20.8·49-s − 5.00·61-s + 16·67-s − 13.8·73-s + 15.0·79-s + 33.3·91-s − 13.8·97-s − 16.5·103-s − 13.2·109-s + ⋯ |
| L(s) = 1 | − 1.99·7-s − 1.75·13-s + 0.794·19-s − 25-s − 0.811·31-s − 1.96·37-s − 1.58·43-s + 2.97·49-s − 0.640·61-s + 1.95·67-s − 1.62·73-s + 1.69·79-s + 3.49·91-s − 1.40·97-s − 1.63·103-s − 1.27·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4444405320\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4444405320\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 + 5.27T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 6.31T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 3.46T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 4.52T + 31T^{2} \) |
| 37 | \( 1 + 11.9T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 5.00T + 61T^{2} \) |
| 67 | \( 1 - 16T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 13.8T + 73T^{2} \) |
| 79 | \( 1 - 15.0T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 13.8T + 97T^{2} \) |
| 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
−7.51575601322194832298818004622, −6.95779178977527423257721534179, −6.56008219693762613061564575636, −5.56268289066454963226288484590, −5.17070303823346492077175950706, −4.04111241080636171150165958289, −3.36689300676674853445385804829, −2.77819491098741114169580916619, −1.85888278633763836280341812485, −0.29900260853018154502507582581,
0.29900260853018154502507582581, 1.85888278633763836280341812485, 2.77819491098741114169580916619, 3.36689300676674853445385804829, 4.04111241080636171150165958289, 5.17070303823346492077175950706, 5.56268289066454963226288484590, 6.56008219693762613061564575636, 6.95779178977527423257721534179, 7.51575601322194832298818004622
