L(s) = 1 | − 3·7-s + 5·11-s − 4·13-s + 8·17-s + 2·19-s − 2·23-s + 6·29-s − 7·31-s + 6·37-s − 6·41-s + 2·43-s − 6·47-s + 2·49-s − 5·53-s − 4·59-s − 8·61-s + 10·67-s − 8·71-s − 73-s − 15·77-s + 16·79-s + 11·83-s + 6·89-s + 12·91-s + 97-s + 9·101-s − 4·103-s + ⋯ |
L(s) = 1 | − 1.13·7-s + 1.50·11-s − 1.10·13-s + 1.94·17-s + 0.458·19-s − 0.417·23-s + 1.11·29-s − 1.25·31-s + 0.986·37-s − 0.937·41-s + 0.304·43-s − 0.875·47-s + 2/7·49-s − 0.686·53-s − 0.520·59-s − 1.02·61-s + 1.22·67-s − 0.949·71-s − 0.117·73-s − 1.70·77-s + 1.80·79-s + 1.20·83-s + 0.635·89-s + 1.25·91-s + 0.101·97-s + 0.895·101-s − 0.394·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.757679663\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.757679663\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−8.042172775086403233459481142387, −7.44894336279083440986726286707, −6.67242977020242524932904531217, −6.15101975632216486800918699194, −5.33844637998305731663743423872, −4.46983348828074326073769615982, −3.45826646873988991137889754792, −3.12728057834934700153878696444, −1.81622585189479262412470937833, −0.72305262574385043562702616235,
0.72305262574385043562702616235, 1.81622585189479262412470937833, 3.12728057834934700153878696444, 3.45826646873988991137889754792, 4.46983348828074326073769615982, 5.33844637998305731663743423872, 6.15101975632216486800918699194, 6.67242977020242524932904531217, 7.44894336279083440986726286707, 8.042172775086403233459481142387