| L(s) = 1 | − 2-s + 4-s − 4·7-s − 8-s − 4·13-s + 4·14-s + 16-s − 6·17-s + 19-s + 6·23-s − 5·25-s + 4·26-s − 4·28-s − 6·29-s + 2·31-s − 32-s + 6·34-s − 4·37-s − 38-s − 6·41-s − 4·43-s − 6·46-s − 6·47-s + 9·49-s + 5·50-s − 4·52-s − 6·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.51·7-s − 0.353·8-s − 1.10·13-s + 1.06·14-s + 1/4·16-s − 1.45·17-s + 0.229·19-s + 1.25·23-s − 25-s + 0.784·26-s − 0.755·28-s − 1.11·29-s + 0.359·31-s − 0.176·32-s + 1.02·34-s − 0.657·37-s − 0.162·38-s − 0.937·41-s − 0.609·43-s − 0.884·46-s − 0.875·47-s + 9/7·49-s + 0.707·50-s − 0.554·52-s − 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−10.95189135954596859968721862656, −9.814927382770120407835166865620, −9.465004478872581430365849790992, −8.381778434403041270610948676969, −7.06756557503064388001592546481, −6.59547283064248335138694127148, −5.21083603966480863402338802043, −3.58677867797817326821260752741, −2.34092707995523044468378525917, 0,
2.34092707995523044468378525917, 3.58677867797817326821260752741, 5.21083603966480863402338802043, 6.59547283064248335138694127148, 7.06756557503064388001592546481, 8.381778434403041270610948676969, 9.465004478872581430365849790992, 9.814927382770120407835166865620, 10.95189135954596859968721862656
