L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 7-s − 8-s + 9-s + 10-s + 12-s − 13-s + 14-s − 15-s + 16-s − 3·17-s − 18-s + 4·19-s − 20-s − 21-s + 8·23-s − 24-s + 25-s + 26-s + 27-s − 28-s + 29-s + 30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s − 0.277·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.727·17-s − 0.235·18-s + 0.917·19-s − 0.223·20-s − 0.218·21-s + 1.66·23-s − 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s − 0.188·28-s + 0.185·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 - 5 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.72657697436532, −14.58431675065897, −13.85833515280789, −13.17449546532540, −12.88288711689914, −12.24139771790649, −11.66674085678906, −11.15105098269042, −10.68267908154381, −10.07685549739942, −9.435192277938335, −9.159664292661574, −8.595386948000041, −8.036467846307618, −7.431350709247289, −7.056813246108136, −6.510836756706039, −5.810325670829045, −4.932985196018099, −4.538844205989013, −3.576420286642627, −3.083918867689546, −2.592377042031520, −1.669461363682274, −0.9480301682485572, 0,
0.9480301682485572, 1.669461363682274, 2.592377042031520, 3.083918867689546, 3.576420286642627, 4.538844205989013, 4.932985196018099, 5.810325670829045, 6.510836756706039, 7.056813246108136, 7.431350709247289, 8.036467846307618, 8.595386948000041, 9.159664292661574, 9.435192277938335, 10.07685549739942, 10.68267908154381, 11.15105098269042, 11.66674085678906, 12.24139771790649, 12.88288711689914, 13.17449546532540, 13.85833515280789, 14.58431675065897, 14.72657697436532