| L(s) = 1 | − 2·4-s − 3·5-s − 7-s − 3·11-s − 4·13-s + 4·16-s + 3·17-s + 19-s + 6·20-s + 4·25-s + 2·28-s − 6·29-s − 4·31-s + 3·35-s + 2·37-s + 6·41-s − 43-s + 6·44-s + 3·47-s − 6·49-s + 8·52-s − 12·53-s + 9·55-s + 6·59-s − 61-s − 8·64-s + 12·65-s + ⋯ |
| L(s) = 1 | − 4-s − 1.34·5-s − 0.377·7-s − 0.904·11-s − 1.10·13-s + 16-s + 0.727·17-s + 0.229·19-s + 1.34·20-s + 4/5·25-s + 0.377·28-s − 1.11·29-s − 0.718·31-s + 0.507·35-s + 0.328·37-s + 0.937·41-s − 0.152·43-s + 0.904·44-s + 0.437·47-s − 6/7·49-s + 1.10·52-s − 1.64·53-s + 1.21·55-s + 0.781·59-s − 0.128·61-s − 64-s + 1.48·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−12.45987383937424874038080001140, −11.37237044309941020825023181952, −10.15333178148614777799832547132, −9.238767264318381906771348588364, −7.968751144610877316457317025190, −7.43141403312000422638829977686, −5.51097905283736056328679977781, −4.41466206651142691425988858089, −3.23857552173567549926694025031, 0,
3.23857552173567549926694025031, 4.41466206651142691425988858089, 5.51097905283736056328679977781, 7.43141403312000422638829977686, 7.968751144610877316457317025190, 9.238767264318381906771348588364, 10.15333178148614777799832547132, 11.37237044309941020825023181952, 12.45987383937424874038080001140
