| L(s) = 1 | − 2·2-s + 4·4-s + 3·5-s − 10·7-s − 8·8-s − 6·10-s − 11·11-s − 16·13-s + 20·14-s + 16·16-s − 42·17-s + 116·19-s + 12·20-s + 22·22-s − 189·23-s − 116·25-s + 32·26-s − 40·28-s + 120·29-s − 163·31-s − 32·32-s + 84·34-s − 30·35-s − 409·37-s − 232·38-s − 24·40-s − 468·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.268·5-s − 0.539·7-s − 0.353·8-s − 0.189·10-s − 0.301·11-s − 0.341·13-s + 0.381·14-s + 1/4·16-s − 0.599·17-s + 1.40·19-s + 0.134·20-s + 0.213·22-s − 1.71·23-s − 0.927·25-s + 0.241·26-s − 0.269·28-s + 0.768·29-s − 0.944·31-s − 0.176·32-s + 0.423·34-s − 0.144·35-s − 1.81·37-s − 0.990·38-s − 0.0948·40-s − 1.78·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p T \) |
| 3 | \( 1 \) |
| 11 | \( 1 + p T \) |
| good | 5 | \( 1 - 3 T + p^{3} T^{2} \) |
| 7 | \( 1 + 10 T + p^{3} T^{2} \) |
| 13 | \( 1 + 16 T + p^{3} T^{2} \) |
| 17 | \( 1 + 42 T + p^{3} T^{2} \) |
| 19 | \( 1 - 116 T + p^{3} T^{2} \) |
| 23 | \( 1 + 189 T + p^{3} T^{2} \) |
| 29 | \( 1 - 120 T + p^{3} T^{2} \) |
| 31 | \( 1 + 163 T + p^{3} T^{2} \) |
| 37 | \( 1 + 409 T + p^{3} T^{2} \) |
| 41 | \( 1 + 468 T + p^{3} T^{2} \) |
| 43 | \( 1 - 110 T + p^{3} T^{2} \) |
| 47 | \( 1 + 144 T + p^{3} T^{2} \) |
| 53 | \( 1 + 90 T + p^{3} T^{2} \) |
| 59 | \( 1 - 453 T + p^{3} T^{2} \) |
| 61 | \( 1 - 20 T + p^{3} T^{2} \) |
| 67 | \( 1 + 97 T + p^{3} T^{2} \) |
| 71 | \( 1 - 465 T + p^{3} T^{2} \) |
| 73 | \( 1 - 848 T + p^{3} T^{2} \) |
| 79 | \( 1 + 742 T + p^{3} T^{2} \) |
| 83 | \( 1 + 438 T + p^{3} T^{2} \) |
| 89 | \( 1 - 273 T + p^{3} T^{2} \) |
| 97 | \( 1 - 761 T + p^{3} T^{2} \) |
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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
−11.51850872074078928558636180726, −10.21496328323924811104255443804, −9.721469382707978309685127074894, −8.575634539377010136503479721891, −7.51547574208879058524970865627, −6.46681109379557858265751636856, −5.28377124673325022708646604567, −3.48237143684882026483385109418, −1.96101718590577458765556343854, 0,
1.96101718590577458765556343854, 3.48237143684882026483385109418, 5.28377124673325022708646604567, 6.46681109379557858265751636856, 7.51547574208879058524970865627, 8.575634539377010136503479721891, 9.721469382707978309685127074894, 10.21496328323924811104255443804, 11.51850872074078928558636180726
