| L(s) = 1 | − 3-s − 2·9-s − 5·11-s − 7·13-s + 3·17-s − 2·19-s − 8·23-s + 5·27-s − 5·29-s − 10·31-s + 5·33-s − 4·37-s + 7·39-s − 6·41-s − 2·43-s + 7·47-s − 3·51-s + 10·53-s + 2·57-s − 10·59-s − 12·61-s + 2·67-s + 8·69-s + 2·73-s − 7·79-s + 81-s − 4·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 2/3·9-s − 1.50·11-s − 1.94·13-s + 0.727·17-s − 0.458·19-s − 1.66·23-s + 0.962·27-s − 0.928·29-s − 1.79·31-s + 0.870·33-s − 0.657·37-s + 1.12·39-s − 0.937·41-s − 0.304·43-s + 1.02·47-s − 0.420·51-s + 1.37·53-s + 0.264·57-s − 1.30·59-s − 1.53·61-s + 0.244·67-s + 0.963·69-s + 0.234·73-s − 0.787·79-s + 1/9·81-s − 0.439·83-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 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−7.23205026722869649248206341197, −6.10949494686391061322295246692, −5.45689039965053936722434021658, −5.21926221949772257635502263037, −4.33314292388597968962225899622, −3.33687065595145403163050956139, −2.52302369004930905815314455290, −1.88639777977204937668456302719, 0, 0,
1.88639777977204937668456302719, 2.52302369004930905815314455290, 3.33687065595145403163050956139, 4.33314292388597968962225899622, 5.21926221949772257635502263037, 5.45689039965053936722434021658, 6.10949494686391061322295246692, 7.23205026722869649248206341197
