| L(s) = 1 | − 3·9-s − 4·11-s + 2·13-s − 6·17-s − 8·19-s + 6·29-s − 8·31-s + 2·37-s − 2·41-s + 4·43-s − 8·47-s − 6·53-s + 6·61-s + 4·67-s − 8·71-s + 10·73-s + 16·79-s + 9·81-s + 8·83-s + 6·89-s − 6·97-s + 12·99-s − 2·101-s − 16·103-s + 12·107-s − 10·109-s − 2·113-s + ⋯ |
| L(s) = 1 | − 9-s − 1.20·11-s + 0.554·13-s − 1.45·17-s − 1.83·19-s + 1.11·29-s − 1.43·31-s + 0.328·37-s − 0.312·41-s + 0.609·43-s − 1.16·47-s − 0.824·53-s + 0.768·61-s + 0.488·67-s − 0.949·71-s + 1.17·73-s + 1.80·79-s + 81-s + 0.878·83-s + 0.635·89-s − 0.609·97-s + 1.20·99-s − 0.199·101-s − 1.57·103-s + 1.16·107-s − 0.957·109-s − 0.188·113-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
| \(L(1)\) |
\(\approx\) |
\(0.7762611532\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7762611532\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−7.937898621674792722572145136667, −6.76235629510354284957581403084, −6.41907891594699397334480521014, −5.62879135136296003772646624556, −4.93508453663962019092646771617, −4.25008554591562906210623936786, −3.36242490236447351131250672642, −2.49230455256797964050759319543, −1.97525845344195001068851519117, −0.38844737372321138122894515881,
0.38844737372321138122894515881, 1.97525845344195001068851519117, 2.49230455256797964050759319543, 3.36242490236447351131250672642, 4.25008554591562906210623936786, 4.93508453663962019092646771617, 5.62879135136296003772646624556, 6.41907891594699397334480521014, 6.76235629510354284957581403084, 7.937898621674792722572145136667
