| L(s) = 1 | + 2.01·2-s + 3-s + 2.05·4-s − 0.0496·5-s + 2.01·6-s + 7-s + 0.113·8-s + 9-s − 0.100·10-s − 3.98·11-s + 2.05·12-s − 3.05·13-s + 2.01·14-s − 0.0496·15-s − 3.88·16-s − 5.13·17-s + 2.01·18-s + 5.91·19-s − 0.102·20-s + 21-s − 8.02·22-s + 2.48·23-s + 0.113·24-s − 4.99·25-s − 6.15·26-s + 27-s + 2.05·28-s + ⋯ |
| L(s) = 1 | + 1.42·2-s + 0.577·3-s + 1.02·4-s − 0.0222·5-s + 0.822·6-s + 0.377·7-s + 0.0400·8-s + 0.333·9-s − 0.0316·10-s − 1.20·11-s + 0.593·12-s − 0.847·13-s + 0.538·14-s − 0.0128·15-s − 0.971·16-s − 1.24·17-s + 0.474·18-s + 1.35·19-s − 0.0228·20-s + 0.218·21-s − 1.71·22-s + 0.517·23-s + 0.0231·24-s − 0.999·25-s − 1.20·26-s + 0.192·27-s + 0.388·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 - 2.01T + 2T^{2} \) |
| 5 | \( 1 + 0.0496T + 5T^{2} \) |
| 11 | \( 1 + 3.98T + 11T^{2} \) |
| 13 | \( 1 + 3.05T + 13T^{2} \) |
| 17 | \( 1 + 5.13T + 17T^{2} \) |
| 19 | \( 1 - 5.91T + 19T^{2} \) |
| 23 | \( 1 - 2.48T + 23T^{2} \) |
| 29 | \( 1 + 2.58T + 29T^{2} \) |
| 31 | \( 1 - 0.304T + 31T^{2} \) |
| 37 | \( 1 - 2.79T + 37T^{2} \) |
| 41 | \( 1 + 4.47T + 41T^{2} \) |
| 43 | \( 1 - 2.27T + 43T^{2} \) |
| 47 | \( 1 + 6.05T + 47T^{2} \) |
| 53 | \( 1 - 1.33T + 53T^{2} \) |
| 59 | \( 1 + 0.422T + 59T^{2} \) |
| 61 | \( 1 - 9.51T + 61T^{2} \) |
| 67 | \( 1 - 6.58T + 67T^{2} \) |
| 71 | \( 1 + 13.0T + 71T^{2} \) |
| 73 | \( 1 + 14.0T + 73T^{2} \) |
| 79 | \( 1 + 13.1T + 79T^{2} \) |
| 83 | \( 1 + 4.38T + 83T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
| 97 | \( 1 + 18.6T + 97T^{2} \) |
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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
−7.27286216058848715286401147648, −6.87500897807669569723053011680, −5.70571529038674067333500293681, −5.34610780649388753725030625222, −4.58628249520980678029059081810, −4.07457328915836180100903382392, −3.02768987620577853429201327749, −2.64908973989196566680069925980, −1.75537617309958849811229886336, 0,
1.75537617309958849811229886336, 2.64908973989196566680069925980, 3.02768987620577853429201327749, 4.07457328915836180100903382392, 4.58628249520980678029059081810, 5.34610780649388753725030625222, 5.70571529038674067333500293681, 6.87500897807669569723053011680, 7.27286216058848715286401147648
