| L(s) = 1 | − 3·3-s + 6·9-s + 3·11-s − 13-s + 5·17-s + 8·19-s + 2·23-s − 9·27-s − 29-s + 2·31-s − 9·33-s + 10·37-s + 3·39-s + 6·41-s − 4·43-s − 11·47-s − 15·51-s + 6·53-s − 24·57-s + 10·59-s − 10·67-s − 6·69-s + 10·73-s − 7·79-s + 9·81-s − 12·83-s + 3·87-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 2·9-s + 0.904·11-s − 0.277·13-s + 1.21·17-s + 1.83·19-s + 0.417·23-s − 1.73·27-s − 0.185·29-s + 0.359·31-s − 1.56·33-s + 1.64·37-s + 0.480·39-s + 0.937·41-s − 0.609·43-s − 1.60·47-s − 2.10·51-s + 0.824·53-s − 3.17·57-s + 1.30·59-s − 1.22·67-s − 0.722·69-s + 1.17·73-s − 0.787·79-s + 81-s − 1.31·83-s + 0.321·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.281155651\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.281155651\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−8.025265747999977960943014516750, −7.33030722314022194729244691981, −6.74218839014611467854111618741, −5.94765260083904574052969903527, −5.45345857683019602647292407838, −4.78750438942502800502515301013, −3.94906456025985550691815848792, −2.97644214347808733552832716710, −1.39504087942516283792497353190, −0.78170268943067997696658926308,
0.78170268943067997696658926308, 1.39504087942516283792497353190, 2.97644214347808733552832716710, 3.94906456025985550691815848792, 4.78750438942502800502515301013, 5.45345857683019602647292407838, 5.94765260083904574052969903527, 6.74218839014611467854111618741, 7.33030722314022194729244691981, 8.025265747999977960943014516750
