L(s) = 1 | − 2·3-s + 9-s + 4·11-s + 2·13-s + 3·17-s − 3·23-s + 4·27-s − 6·29-s + 9·31-s − 8·33-s − 4·39-s + 5·41-s + 6·43-s − 9·47-s − 6·51-s + 6·53-s + 8·59-s + 8·61-s − 14·67-s + 6·69-s + 11·71-s + 2·73-s + 9·79-s − 11·81-s + 6·83-s + 12·87-s + 11·89-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 1/3·9-s + 1.20·11-s + 0.554·13-s + 0.727·17-s − 0.625·23-s + 0.769·27-s − 1.11·29-s + 1.61·31-s − 1.39·33-s − 0.640·39-s + 0.780·41-s + 0.914·43-s − 1.31·47-s − 0.840·51-s + 0.824·53-s + 1.04·59-s + 1.02·61-s − 1.71·67-s + 0.722·69-s + 1.30·71-s + 0.234·73-s + 1.01·79-s − 1.22·81-s + 0.658·83-s + 1.28·87-s + 1.16·89-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
\(L(1)\) |
\(\approx\) |
\(1.476701388\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.476701388\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 - 11 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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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
−7.64078789104580501144312381371, −6.63843906510412755080474917666, −6.37648843147059525413329210534, −5.66744488388680810756011445118, −5.10125760386806139583811234690, −4.16259031967932193468344256228, −3.66553005428872706532894780818, −2.56770843961104187995913591259, −1.40954196697893242718454203563, −0.67728529679948085997858779241,
0.67728529679948085997858779241, 1.40954196697893242718454203563, 2.56770843961104187995913591259, 3.66553005428872706532894780818, 4.16259031967932193468344256228, 5.10125760386806139583811234690, 5.66744488388680810756011445118, 6.37648843147059525413329210534, 6.63843906510412755080474917666, 7.64078789104580501144312381371