| L(s) = 1 | − 3-s − 5-s − 7-s − 2·9-s − 5·11-s + 13-s + 15-s − 2·17-s − 4·19-s + 21-s − 4·23-s − 4·25-s + 5·27-s + 5·31-s + 5·33-s + 35-s + 2·37-s − 39-s − 43-s + 2·45-s − 7·47-s + 49-s + 2·51-s + 9·53-s + 5·55-s + 4·57-s − 10·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s − 2/3·9-s − 1.50·11-s + 0.277·13-s + 0.258·15-s − 0.485·17-s − 0.917·19-s + 0.218·21-s − 0.834·23-s − 4/5·25-s + 0.962·27-s + 0.898·31-s + 0.870·33-s + 0.169·35-s + 0.328·37-s − 0.160·39-s − 0.152·43-s + 0.298·45-s − 1.02·47-s + 1/7·49-s + 0.280·51-s + 1.23·53-s + 0.674·55-s + 0.529·57-s − 1.30·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 94192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 94192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
| 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
−13.82681377002951, −13.59133248419979, −13.09652660066064, −12.50429778951873, −12.10450012961075, −11.59344653506697, −11.12126656027961, −10.59516306390469, −10.32469283138741, −9.677938698962030, −9.074126534772762, −8.379597197103356, −8.068081301320696, −7.700605398230153, −6.800679058088548, −6.426731814449977, −5.870317740434276, −5.403333669389322, −4.817277410963066, −4.215359515475992, −3.658499461809857, −2.796957145726995, −2.510752778959778, −1.653619299502078, −0.5072825738012377, 0,
0.5072825738012377, 1.653619299502078, 2.510752778959778, 2.796957145726995, 3.658499461809857, 4.215359515475992, 4.817277410963066, 5.403333669389322, 5.870317740434276, 6.426731814449977, 6.800679058088548, 7.700605398230153, 8.068081301320696, 8.379597197103356, 9.074126534772762, 9.677938698962030, 10.32469283138741, 10.59516306390469, 11.12126656027961, 11.59344653506697, 12.10450012961075, 12.50429778951873, 13.09652660066064, 13.59133248419979, 13.82681377002951
