| L(s) = 1 | − 2·2-s + 3-s + 2·4-s + 5-s − 2·6-s + 2·7-s − 2·9-s − 2·10-s − 11-s + 2·12-s − 4·13-s − 4·14-s + 15-s − 4·16-s + 2·17-s + 4·18-s + 2·20-s + 2·21-s + 2·22-s − 23-s − 4·25-s + 8·26-s − 5·27-s + 4·28-s − 2·30-s + 7·31-s + 8·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s + 0.447·5-s − 0.816·6-s + 0.755·7-s − 2/3·9-s − 0.632·10-s − 0.301·11-s + 0.577·12-s − 1.10·13-s − 1.06·14-s + 0.258·15-s − 16-s + 0.485·17-s + 0.942·18-s + 0.447·20-s + 0.436·21-s + 0.426·22-s − 0.208·23-s − 4/5·25-s + 1.56·26-s − 0.962·27-s + 0.755·28-s − 0.365·30-s + 1.25·31-s + 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18491 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18491 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + T \) |
| 41 | \( 1 \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−16.24456738947274, −15.52568747389412, −14.96445316063292, −14.42659003422698, −13.97014246235524, −13.43718677385936, −12.77460952787535, −11.80532652020216, −11.57979595676345, −10.91836655063080, −10.10892538584067, −9.807053443777563, −9.426910093720351, −8.533161498536632, −8.168656713206125, −7.896723878259049, −7.124403448999743, −6.497865314949136, −5.558489055892140, −5.038060269650380, −4.257842775528266, −3.238250867554864, −2.347520168887076, −2.041578279483596, −1.025745927093472, 0,
1.025745927093472, 2.041578279483596, 2.347520168887076, 3.238250867554864, 4.257842775528266, 5.038060269650380, 5.558489055892140, 6.497865314949136, 7.124403448999743, 7.896723878259049, 8.168656713206125, 8.533161498536632, 9.426910093720351, 9.807053443777563, 10.10892538584067, 10.91836655063080, 11.57979595676345, 11.80532652020216, 12.77460952787535, 13.43718677385936, 13.97014246235524, 14.42659003422698, 14.96445316063292, 15.52568747389412, 16.24456738947274
