L(s) = 1 | + 7-s − 3·9-s + 11-s − 2·13-s − 4·17-s − 2·19-s − 5·23-s + 29-s − 2·31-s − 3·37-s + 12·41-s − 11·43-s − 2·47-s + 49-s − 6·53-s − 10·59-s + 4·61-s − 3·63-s − 67-s − 3·71-s + 77-s − 9·79-s + 9·81-s + 2·83-s − 6·89-s − 2·91-s − 14·97-s + ⋯ |
L(s) = 1 | + 0.377·7-s − 9-s + 0.301·11-s − 0.554·13-s − 0.970·17-s − 0.458·19-s − 1.04·23-s + 0.185·29-s − 0.359·31-s − 0.493·37-s + 1.87·41-s − 1.67·43-s − 0.291·47-s + 1/7·49-s − 0.824·53-s − 1.30·59-s + 0.512·61-s − 0.377·63-s − 0.122·67-s − 0.356·71-s + 0.113·77-s − 1.01·79-s + 81-s + 0.219·83-s − 0.635·89-s − 0.209·91-s − 1.42·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−9.063732721407264098804255752994, −8.410319456039944378168425749912, −7.64853346454933513923745764234, −6.62385461300390207913727757851, −5.89185191436335680897716961439, −4.92280621580036592880794499259, −4.06299815122178043432589638506, −2.85830357112156475553686029481, −1.85727000132990006956153760338, 0,
1.85727000132990006956153760338, 2.85830357112156475553686029481, 4.06299815122178043432589638506, 4.92280621580036592880794499259, 5.89185191436335680897716961439, 6.62385461300390207913727757851, 7.64853346454933513923745764234, 8.410319456039944378168425749912, 9.063732721407264098804255752994