| L(s) = 1 | − 2·3-s + 9-s − 4·11-s − 4·13-s + 4·19-s − 2·23-s + 4·27-s + 2·29-s + 8·33-s + 4·37-s + 8·39-s − 2·41-s − 6·43-s + 6·47-s − 4·53-s − 8·57-s + 12·59-s + 10·61-s + 14·67-s + 4·69-s + 8·71-s − 8·73-s + 16·79-s − 11·81-s − 2·83-s − 4·87-s − 6·89-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/3·9-s − 1.20·11-s − 1.10·13-s + 0.917·19-s − 0.417·23-s + 0.769·27-s + 0.371·29-s + 1.39·33-s + 0.657·37-s + 1.28·39-s − 0.312·41-s − 0.914·43-s + 0.875·47-s − 0.549·53-s − 1.05·57-s + 1.56·59-s + 1.28·61-s + 1.71·67-s + 0.481·69-s + 0.949·71-s − 0.936·73-s + 1.80·79-s − 1.22·81-s − 0.219·83-s − 0.428·87-s − 0.635·89-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−7.14921286604979326658019660163, −6.70704043705589798939692431076, −5.77444390103934092747591743437, −5.25454667818013035065281812785, −4.92094480034658234190771257579, −3.94230043837437753170237232424, −2.88727452627385173253869196408, −2.24846064180628551899679631269, −0.903711169250688522174204642296, 0,
0.903711169250688522174204642296, 2.24846064180628551899679631269, 2.88727452627385173253869196408, 3.94230043837437753170237232424, 4.92094480034658234190771257579, 5.25454667818013035065281812785, 5.77444390103934092747591743437, 6.70704043705589798939692431076, 7.14921286604979326658019660163
