| L(s) = 1 | − 3-s − 5-s − 7-s − 2·9-s − 3·11-s + 3·13-s + 15-s + 5·17-s + 21-s + 8·23-s + 25-s + 5·27-s − 29-s + 4·31-s + 3·33-s + 35-s − 8·37-s − 3·39-s + 4·41-s + 12·43-s + 2·45-s + 3·47-s + 49-s − 5·51-s + 3·55-s + 2·59-s − 10·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s − 2/3·9-s − 0.904·11-s + 0.832·13-s + 0.258·15-s + 1.21·17-s + 0.218·21-s + 1.66·23-s + 1/5·25-s + 0.962·27-s − 0.185·29-s + 0.718·31-s + 0.522·33-s + 0.169·35-s − 1.31·37-s − 0.480·39-s + 0.624·41-s + 1.82·43-s + 0.298·45-s + 0.437·47-s + 1/7·49-s − 0.700·51-s + 0.404·55-s + 0.260·59-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−13.21071032864990, −12.67901830513986, −12.38300883069695, −11.88388578562627, −11.41001231019510, −10.85736947619408, −10.60257056880807, −10.25949755045542, −9.398849721151964, −9.030068575763933, −8.605222190557437, −7.949138273307561, −7.615991008860932, −7.072625778045800, −6.486111493144854, −5.902641827209367, −5.627997647986815, −5.009522446528103, −4.636597914269073, −3.775704062656327, −3.307106137863271, −2.871472099820687, −2.287658706984341, −1.171379094476316, −0.8116259708938053, 0,
0.8116259708938053, 1.171379094476316, 2.287658706984341, 2.871472099820687, 3.307106137863271, 3.775704062656327, 4.636597914269073, 5.009522446528103, 5.627997647986815, 5.902641827209367, 6.486111493144854, 7.072625778045800, 7.615991008860932, 7.949138273307561, 8.605222190557437, 9.030068575763933, 9.398849721151964, 10.25949755045542, 10.60257056880807, 10.85736947619408, 11.41001231019510, 11.88388578562627, 12.38300883069695, 12.67901830513986, 13.21071032864990
