L(s) = 1 | − 0.585·5-s + 1.41·7-s + 5.65·11-s + 5.65·13-s + 2.24·17-s + 8.24·19-s + 23-s − 4.65·25-s − 8.82·29-s + 1.17·31-s − 0.828·35-s − 3.17·37-s + 2·41-s + 5.41·43-s − 10.4·47-s − 5·49-s − 4.58·53-s − 3.31·55-s − 8.82·59-s + 3.17·61-s − 3.31·65-s + 7.75·67-s + 11.3·71-s + 0.343·73-s + 8.00·77-s + 7.07·79-s − 9.65·83-s + ⋯ |
L(s) = 1 | − 0.261·5-s + 0.534·7-s + 1.70·11-s + 1.56·13-s + 0.543·17-s + 1.89·19-s + 0.208·23-s − 0.931·25-s − 1.63·29-s + 0.210·31-s − 0.140·35-s − 0.521·37-s + 0.312·41-s + 0.825·43-s − 1.52·47-s − 0.714·49-s − 0.629·53-s − 0.446·55-s − 1.14·59-s + 0.406·61-s − 0.411·65-s + 0.947·67-s + 1.34·71-s + 0.0401·73-s + 0.911·77-s + 0.795·79-s − 1.05·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.470277500\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.470277500\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 + 0.585T + 5T^{2} \) |
| 7 | \( 1 - 1.41T + 7T^{2} \) |
| 11 | \( 1 - 5.65T + 11T^{2} \) |
| 13 | \( 1 - 5.65T + 13T^{2} \) |
| 17 | \( 1 - 2.24T + 17T^{2} \) |
| 19 | \( 1 - 8.24T + 19T^{2} \) |
| 29 | \( 1 + 8.82T + 29T^{2} \) |
| 31 | \( 1 - 1.17T + 31T^{2} \) |
| 37 | \( 1 + 3.17T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 5.41T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 + 4.58T + 53T^{2} \) |
| 59 | \( 1 + 8.82T + 59T^{2} \) |
| 61 | \( 1 - 3.17T + 61T^{2} \) |
| 67 | \( 1 - 7.75T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 - 0.343T + 73T^{2} \) |
| 79 | \( 1 - 7.07T + 79T^{2} \) |
| 83 | \( 1 + 9.65T + 83T^{2} \) |
| 89 | \( 1 + 13.0T + 89T^{2} \) |
| 97 | \( 1 - 16.1T + 97T^{2} \) |
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
−8.628246848744326063165461632256, −7.87676131110686305246231121658, −7.24077171331293341533102914586, −6.30127344626432102642594393773, −5.71302443264277294786918145121, −4.77245546203014375723983177088, −3.62885930813032906937200412125, −3.50606774770284658284244815164, −1.72739403628376644971139518041, −1.06812552608721381515313313957,
1.06812552608721381515313313957, 1.72739403628376644971139518041, 3.50606774770284658284244815164, 3.62885930813032906937200412125, 4.77245546203014375723983177088, 5.71302443264277294786918145121, 6.30127344626432102642594393773, 7.24077171331293341533102914586, 7.87676131110686305246231121658, 8.628246848744326063165461632256