L(s) = 1 | + 3-s + 7·5-s − 26·9-s − 35·11-s + 66·13-s + 7·15-s + 59·17-s − 137·19-s + 7·23-s − 76·25-s − 53·27-s + 106·29-s − 75·31-s − 35·33-s + 11·37-s + 66·39-s − 498·41-s − 260·43-s − 182·45-s + 171·47-s + 59·51-s − 417·53-s − 245·55-s − 137·57-s + 17·59-s + 51·61-s + 462·65-s + ⋯ |
L(s) = 1 | + 0.192·3-s + 0.626·5-s − 0.962·9-s − 0.959·11-s + 1.40·13-s + 0.120·15-s + 0.841·17-s − 1.65·19-s + 0.0634·23-s − 0.607·25-s − 0.377·27-s + 0.678·29-s − 0.434·31-s − 0.184·33-s + 0.0488·37-s + 0.270·39-s − 1.89·41-s − 0.922·43-s − 0.602·45-s + 0.530·47-s + 0.161·51-s − 1.08·53-s − 0.600·55-s − 0.318·57-s + 0.0375·59-s + 0.107·61-s + 0.881·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - T + p^{3} T^{2} \) |
| 5 | \( 1 - 7 T + p^{3} T^{2} \) |
| 11 | \( 1 + 35 T + p^{3} T^{2} \) |
| 13 | \( 1 - 66 T + p^{3} T^{2} \) |
| 17 | \( 1 - 59 T + p^{3} T^{2} \) |
| 19 | \( 1 + 137 T + p^{3} T^{2} \) |
| 23 | \( 1 - 7 T + p^{3} T^{2} \) |
| 29 | \( 1 - 106 T + p^{3} T^{2} \) |
| 31 | \( 1 + 75 T + p^{3} T^{2} \) |
| 37 | \( 1 - 11 T + p^{3} T^{2} \) |
| 41 | \( 1 + 498 T + p^{3} T^{2} \) |
| 43 | \( 1 + 260 T + p^{3} T^{2} \) |
| 47 | \( 1 - 171 T + p^{3} T^{2} \) |
| 53 | \( 1 + 417 T + p^{3} T^{2} \) |
| 59 | \( 1 - 17 T + p^{3} T^{2} \) |
| 61 | \( 1 - 51 T + p^{3} T^{2} \) |
| 67 | \( 1 + 439 T + p^{3} T^{2} \) |
| 71 | \( 1 - 784 T + p^{3} T^{2} \) |
| 73 | \( 1 - 295 T + p^{3} T^{2} \) |
| 79 | \( 1 - 495 T + p^{3} T^{2} \) |
| 83 | \( 1 + 932 T + p^{3} T^{2} \) |
| 89 | \( 1 + 873 T + p^{3} T^{2} \) |
| 97 | \( 1 + 290 T + p^{3} T^{2} \) |
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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
−9.467028984163928199362895813576, −8.394044462086092177065674788532, −8.165974958671036572500893817969, −6.66982069346941180604386068820, −5.91251067461905381501729832388, −5.17007414542838741455380662210, −3.78499243702856244046725272724, −2.77427379711381541627513026585, −1.65609860242055915501292482839, 0,
1.65609860242055915501292482839, 2.77427379711381541627513026585, 3.78499243702856244046725272724, 5.17007414542838741455380662210, 5.91251067461905381501729832388, 6.66982069346941180604386068820, 8.165974958671036572500893817969, 8.394044462086092177065674788532, 9.467028984163928199362895813576