| L(s) = 1 | + 1.44·3-s − 2.86·7-s − 0.909·9-s + 3.84·11-s + 6.22·13-s − 17-s − 6.62·19-s − 4.14·21-s − 4.51·23-s − 5.65·27-s − 0.658·29-s − 3.49·31-s + 5.56·33-s + 3.34·37-s + 8.99·39-s + 2.04·41-s + 1.29·43-s − 6.22·47-s + 1.21·49-s − 1.44·51-s − 9.92·53-s − 9.57·57-s + 2·59-s − 7.61·61-s + 2.60·63-s − 0.257·67-s − 6.53·69-s + ⋯ |
| L(s) = 1 | + 0.834·3-s − 1.08·7-s − 0.303·9-s + 1.15·11-s + 1.72·13-s − 0.242·17-s − 1.51·19-s − 0.904·21-s − 0.942·23-s − 1.08·27-s − 0.122·29-s − 0.628·31-s + 0.968·33-s + 0.549·37-s + 1.44·39-s + 0.318·41-s + 0.197·43-s − 0.907·47-s + 0.172·49-s − 0.202·51-s − 1.36·53-s − 1.26·57-s + 0.260·59-s − 0.975·61-s + 0.328·63-s − 0.0314·67-s − 0.786·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 1.44T + 3T^{2} \) |
| 7 | \( 1 + 2.86T + 7T^{2} \) |
| 11 | \( 1 - 3.84T + 11T^{2} \) |
| 13 | \( 1 - 6.22T + 13T^{2} \) |
| 19 | \( 1 + 6.62T + 19T^{2} \) |
| 23 | \( 1 + 4.51T + 23T^{2} \) |
| 29 | \( 1 + 0.658T + 29T^{2} \) |
| 31 | \( 1 + 3.49T + 31T^{2} \) |
| 37 | \( 1 - 3.34T + 37T^{2} \) |
| 41 | \( 1 - 2.04T + 41T^{2} \) |
| 43 | \( 1 - 1.29T + 43T^{2} \) |
| 47 | \( 1 + 6.22T + 47T^{2} \) |
| 53 | \( 1 + 9.92T + 53T^{2} \) |
| 59 | \( 1 - 2T + 59T^{2} \) |
| 61 | \( 1 + 7.61T + 61T^{2} \) |
| 67 | \( 1 + 0.257T + 67T^{2} \) |
| 71 | \( 1 + 1.18T + 71T^{2} \) |
| 73 | \( 1 - 3.26T + 73T^{2} \) |
| 79 | \( 1 - 4.99T + 79T^{2} \) |
| 83 | \( 1 + 7.91T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 + 4.08T + 97T^{2} \) |
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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
−7.84790874030701677259824718712, −6.71307364151440690911255199348, −6.25943271982554318149782512107, −5.86676974154447634315902263009, −4.44630577655794166521284187830, −3.72411935238632455625345467930, −3.38084139507656307461373005081, −2.33944454039916738221156041296, −1.46058184425582215520751086161, 0,
1.46058184425582215520751086161, 2.33944454039916738221156041296, 3.38084139507656307461373005081, 3.72411935238632455625345467930, 4.44630577655794166521284187830, 5.86676974154447634315902263009, 6.25943271982554318149782512107, 6.71307364151440690911255199348, 7.84790874030701677259824718712
