| L(s) = 1 | − 3·9-s + 4·13-s + 8·17-s + 10·29-s + 12·37-s − 10·41-s − 7·49-s − 4·53-s + 10·61-s + 16·73-s + 9·81-s − 10·89-s − 8·97-s − 2·101-s + 6·109-s − 16·113-s − 12·117-s + ⋯ |
| L(s) = 1 | − 9-s + 1.10·13-s + 1.94·17-s + 1.85·29-s + 1.97·37-s − 1.56·41-s − 49-s − 0.549·53-s + 1.28·61-s + 1.87·73-s + 81-s − 1.05·89-s − 0.812·97-s − 0.199·101-s + 0.574·109-s − 1.50·113-s − 1.10·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.568356227\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.568356227\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−10.23438680846563734808416701606, −9.481001567833775866135833367998, −8.318458181732681568361082997175, −8.043199677138800978749971209938, −6.64037527171448229645548827286, −5.88600838036285002216361121358, −5.01663774178771706139531826817, −3.66097610339341905494433801925, −2.81202536212746604283359988203, −1.09831956032328018823451631467,
1.09831956032328018823451631467, 2.81202536212746604283359988203, 3.66097610339341905494433801925, 5.01663774178771706139531826817, 5.88600838036285002216361121358, 6.64037527171448229645548827286, 8.043199677138800978749971209938, 8.318458181732681568361082997175, 9.481001567833775866135833367998, 10.23438680846563734808416701606
