| L(s) = 1 | − 2-s + 4-s + 5-s + 3·7-s − 8-s − 10-s − 11-s + 13-s − 3·14-s + 16-s − 17-s + 20-s + 22-s − 23-s − 4·25-s − 26-s + 3·28-s + 3·29-s − 2·31-s − 32-s + 34-s + 3·35-s − 2·37-s − 40-s + 9·41-s − 13·43-s − 44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 1.13·7-s − 0.353·8-s − 0.316·10-s − 0.301·11-s + 0.277·13-s − 0.801·14-s + 1/4·16-s − 0.242·17-s + 0.223·20-s + 0.213·22-s − 0.208·23-s − 4/5·25-s − 0.196·26-s + 0.566·28-s + 0.557·29-s − 0.359·31-s − 0.176·32-s + 0.171·34-s + 0.507·35-s − 0.328·37-s − 0.158·40-s + 1.40·41-s − 1.98·43-s − 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43758 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43758 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 13 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−14.95651108931423, −14.44208458177675, −14.00929216871505, −13.34671910939917, −12.99374075245548, −12.09440985615033, −11.80080042399451, −11.14848235475227, −10.81993998734913, −10.13532016808640, −9.805512697433432, −9.016979861674321, −8.668605880578091, −7.931705295921641, −7.771000292176479, −6.963731471173122, −6.358688125802733, −5.792919692434298, −5.152457071165514, −4.621205683202115, −3.842935405261652, −3.069668883737116, −2.223550148530205, −1.773924235334120, −1.065342282760500, 0,
1.065342282760500, 1.773924235334120, 2.223550148530205, 3.069668883737116, 3.842935405261652, 4.621205683202115, 5.152457071165514, 5.792919692434298, 6.358688125802733, 6.963731471173122, 7.771000292176479, 7.931705295921641, 8.668605880578091, 9.016979861674321, 9.805512697433432, 10.13532016808640, 10.81993998734913, 11.14848235475227, 11.80080042399451, 12.09440985615033, 12.99374075245548, 13.34671910939917, 14.00929216871505, 14.44208458177675, 14.95651108931423
