| L(s) = 1 | + 3·3-s + 6·9-s + 11-s − 4·13-s − 5·17-s − 19-s − 2·23-s + 9·27-s − 8·29-s − 10·31-s + 3·33-s − 6·37-s − 12·39-s + 3·41-s + 4·43-s − 4·47-s − 15·51-s + 6·53-s − 3·57-s − 8·59-s − 10·61-s − 67-s − 6·69-s − 12·71-s − 3·73-s + 6·79-s + 9·81-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 2·9-s + 0.301·11-s − 1.10·13-s − 1.21·17-s − 0.229·19-s − 0.417·23-s + 1.73·27-s − 1.48·29-s − 1.79·31-s + 0.522·33-s − 0.986·37-s − 1.92·39-s + 0.468·41-s + 0.609·43-s − 0.583·47-s − 2.10·51-s + 0.824·53-s − 0.397·57-s − 1.04·59-s − 1.28·61-s − 0.122·67-s − 0.722·69-s − 1.42·71-s − 0.351·73-s + 0.675·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 13 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.44038828043031988692099604732, −7.00655609002137112548765801406, −6.07157794601865689003245913376, −5.09461795485578122034257528770, −4.29468985687568976357913968541, −3.74134866687573472505781216264, −2.98737562200123654773117191887, −2.12121488313202889540939179526, −1.76463276668968228616466455866, 0,
1.76463276668968228616466455866, 2.12121488313202889540939179526, 2.98737562200123654773117191887, 3.74134866687573472505781216264, 4.29468985687568976357913968541, 5.09461795485578122034257528770, 6.07157794601865689003245913376, 7.00655609002137112548765801406, 7.44038828043031988692099604732
