| L(s) = 1 | + 2-s + 4-s − 1.41·5-s + 8-s − 1.41·10-s − 11-s + 5.65·13-s + 16-s + 1.41·17-s − 4.24·19-s − 1.41·20-s − 22-s − 4·23-s − 2.99·25-s + 5.65·26-s − 4.24·31-s + 32-s + 1.41·34-s − 6·37-s − 4.24·38-s − 1.41·40-s + 4.24·41-s − 4·43-s − 44-s − 4·46-s − 1.41·47-s − 2.99·50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.632·5-s + 0.353·8-s − 0.447·10-s − 0.301·11-s + 1.56·13-s + 0.250·16-s + 0.342·17-s − 0.973·19-s − 0.316·20-s − 0.213·22-s − 0.834·23-s − 0.599·25-s + 1.10·26-s − 0.762·31-s + 0.176·32-s + 0.242·34-s − 0.986·37-s − 0.688·38-s − 0.223·40-s + 0.662·41-s − 0.609·43-s − 0.150·44-s − 0.589·46-s − 0.206·47-s − 0.424·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + 1.41T + 5T^{2} \) |
| 13 | \( 1 - 5.65T + 13T^{2} \) |
| 17 | \( 1 - 1.41T + 17T^{2} \) |
| 19 | \( 1 + 4.24T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 4.24T + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 - 4.24T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 1.41T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 2.82T + 59T^{2} \) |
| 61 | \( 1 + 5.65T + 61T^{2} \) |
| 67 | \( 1 + 10T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 4.24T + 73T^{2} \) |
| 79 | \( 1 + 6T + 79T^{2} \) |
| 83 | \( 1 - 15.5T + 83T^{2} \) |
| 89 | \( 1 - 14.1T + 89T^{2} \) |
| 97 | \( 1 + 5.65T + 97T^{2} \) |
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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
−7.37307542589453890728648680099, −6.43891793780202291204829760769, −6.01691563405721255381535357802, −5.30046086165808430594803602271, −4.40060901485890855012077169607, −3.80414807200140265702729408107, −3.33618775732448230238689043757, −2.23883143576820183124192350105, −1.39114037757314821566517643112, 0,
1.39114037757314821566517643112, 2.23883143576820183124192350105, 3.33618775732448230238689043757, 3.80414807200140265702729408107, 4.40060901485890855012077169607, 5.30046086165808430594803602271, 6.01691563405721255381535357802, 6.43891793780202291204829760769, 7.37307542589453890728648680099
