| 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
| \(L(1)\) |
\(\approx\) |
\(1.944924090\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.944924090\) |
| \(L(\frac{3}{2})\) |
|
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
−7.913542347645750857230707406421, −7.07931201570469272427654426567, −6.10662165186134336317729572240, −5.79404981995007243868596718698, −4.96416949659307579796585347069, −3.97225591278326031665092931376, −3.44074252602652586678812083818, −2.56860862632909811506715790751, −1.94831577141934159178180689901, −0.61908236301808100361233337048,
0.61908236301808100361233337048, 1.94831577141934159178180689901, 2.56860862632909811506715790751, 3.44074252602652586678812083818, 3.97225591278326031665092931376, 4.96416949659307579796585347069, 5.79404981995007243868596718698, 6.10662165186134336317729572240, 7.07931201570469272427654426567, 7.913542347645750857230707406421
