| L(s) = 1 | − 2·5-s − 7-s − 6·11-s + 2·13-s + 2·17-s + 19-s + 2·23-s − 25-s − 4·29-s + 2·35-s − 2·37-s − 8·43-s − 6·47-s + 49-s + 4·53-s + 12·55-s + 2·61-s − 4·65-s + 4·67-s − 12·71-s + 6·73-s + 6·77-s + 8·79-s − 10·83-s − 4·85-s + 8·89-s − 2·91-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.377·7-s − 1.80·11-s + 0.554·13-s + 0.485·17-s + 0.229·19-s + 0.417·23-s − 1/5·25-s − 0.742·29-s + 0.338·35-s − 0.328·37-s − 1.21·43-s − 0.875·47-s + 1/7·49-s + 0.549·53-s + 1.61·55-s + 0.256·61-s − 0.496·65-s + 0.488·67-s − 1.42·71-s + 0.702·73-s + 0.683·77-s + 0.900·79-s − 1.09·83-s − 0.433·85-s + 0.847·89-s − 0.209·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9169030383\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9169030383\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.184639677331554303933704650434, −7.61588901075909735518418162462, −7.04778130859601909544589177316, −6.02260122758311729025741163435, −5.33160831109524633136613980263, −4.62328709237547491330504325182, −3.56014806445052128856172122648, −3.10537312677644226539074053260, −1.98071470066691974083222620926, −0.50684573635306972930557790284,
0.50684573635306972930557790284, 1.98071470066691974083222620926, 3.10537312677644226539074053260, 3.56014806445052128856172122648, 4.62328709237547491330504325182, 5.33160831109524633136613980263, 6.02260122758311729025741163435, 7.04778130859601909544589177316, 7.61588901075909735518418162462, 8.184639677331554303933704650434
