L(s) = 1 | − 2-s − 4-s + 2·5-s + 7-s + 3·8-s − 2·10-s + 4·13-s − 14-s − 16-s − 17-s − 7·19-s − 2·20-s + 3·23-s − 25-s − 4·26-s − 28-s + 29-s + 3·31-s − 5·32-s + 34-s + 2·35-s − 2·37-s + 7·38-s + 6·40-s − 7·41-s − 4·43-s − 3·46-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.894·5-s + 0.377·7-s + 1.06·8-s − 0.632·10-s + 1.10·13-s − 0.267·14-s − 1/4·16-s − 0.242·17-s − 1.60·19-s − 0.447·20-s + 0.625·23-s − 1/5·25-s − 0.784·26-s − 0.188·28-s + 0.185·29-s + 0.538·31-s − 0.883·32-s + 0.171·34-s + 0.338·35-s − 0.328·37-s + 1.13·38-s + 0.948·40-s − 1.09·41-s − 0.609·43-s − 0.442·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| 29 | \( 1 - T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 9 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 11 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−13.23385667796210, −12.94072940573909, −12.33027136085052, −11.67500388829160, −11.20178956227052, −10.65993260676769, −10.32431662261623, −10.04240004875332, −9.356639478225032, −8.833068057159767, −8.639943538889671, −8.275605848915277, −7.597132841333213, −7.054482902560258, −6.497354725434956, −6.004821628333061, −5.584980855882989, −4.773967157067315, −4.636884703925662, −3.790453795268227, −3.478394215270333, −2.349447885367241, −2.097702158790839, −1.360384215842542, −0.8653254174222407, 0,
0.8653254174222407, 1.360384215842542, 2.097702158790839, 2.349447885367241, 3.478394215270333, 3.790453795268227, 4.636884703925662, 4.773967157067315, 5.584980855882989, 6.004821628333061, 6.497354725434956, 7.054482902560258, 7.597132841333213, 8.275605848915277, 8.639943538889671, 8.833068057159767, 9.356639478225032, 10.04240004875332, 10.32431662261623, 10.65993260676769, 11.20178956227052, 11.67500388829160, 12.33027136085052, 12.94072940573909, 13.23385667796210