| L(s) = 1 | − 2·3-s − 7-s + 9-s + 4·11-s − 2·13-s − 4·19-s + 2·21-s + 23-s − 5·25-s + 4·27-s + 6·29-s + 2·31-s − 8·33-s + 2·37-s + 4·39-s − 6·41-s + 4·43-s − 10·47-s + 49-s + 6·53-s + 8·57-s − 6·59-s − 4·61-s − 63-s + 4·67-s − 2·69-s + 8·71-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.377·7-s + 1/3·9-s + 1.20·11-s − 0.554·13-s − 0.917·19-s + 0.436·21-s + 0.208·23-s − 25-s + 0.769·27-s + 1.11·29-s + 0.359·31-s − 1.39·33-s + 0.328·37-s + 0.640·39-s − 0.937·41-s + 0.609·43-s − 1.45·47-s + 1/7·49-s + 0.824·53-s + 1.05·57-s − 0.781·59-s − 0.512·61-s − 0.125·63-s + 0.488·67-s − 0.240·69-s + 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
| 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
−17.00012728642254, −16.53300415043381, −15.81176377458403, −15.28097094971232, −14.56470235119990, −14.08811085024020, −13.36478700889056, −12.66846617951719, −12.08958132119920, −11.76384385417602, −11.17644250657891, −10.48263395584682, −9.961574268699181, −9.311957318432471, −8.626621031318346, −7.931194250136415, −7.005831832554492, −6.419420437226026, −6.168875738235536, −5.259664942734222, −4.624579710215873, −3.954606202784269, −3.047022400590179, −2.038864973499305, −0.9891462512697895, 0,
0.9891462512697895, 2.038864973499305, 3.047022400590179, 3.954606202784269, 4.624579710215873, 5.259664942734222, 6.168875738235536, 6.419420437226026, 7.005831832554492, 7.931194250136415, 8.626621031318346, 9.311957318432471, 9.961574268699181, 10.48263395584682, 11.17644250657891, 11.76384385417602, 12.08958132119920, 12.66846617951719, 13.36478700889056, 14.08811085024020, 14.56470235119990, 15.28097094971232, 15.81176377458403, 16.53300415043381, 17.00012728642254
