| L(s) = 1 | + 5-s − 3·7-s − 5·11-s − 4·13-s + 3·17-s − 19-s − 8·23-s − 4·25-s + 2·29-s + 4·31-s − 3·35-s + 10·37-s − 10·41-s + 43-s + 47-s + 2·49-s + 4·53-s − 5·55-s − 6·59-s − 13·61-s − 4·65-s − 12·67-s − 2·71-s + 9·73-s + 15·77-s + 8·79-s + 12·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.13·7-s − 1.50·11-s − 1.10·13-s + 0.727·17-s − 0.229·19-s − 1.66·23-s − 4/5·25-s + 0.371·29-s + 0.718·31-s − 0.507·35-s + 1.64·37-s − 1.56·41-s + 0.152·43-s + 0.145·47-s + 2/7·49-s + 0.549·53-s − 0.674·55-s − 0.781·59-s − 1.66·61-s − 0.496·65-s − 1.46·67-s − 0.237·71-s + 1.05·73-s + 1.70·77-s + 0.900·79-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 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
−9.954901377829594152119131032207, −9.559628331745101664415344236205, −8.146906664648363880391573042940, −7.53862789028003683931489821585, −6.33560598867710411281522365718, −5.65147980657085516848049895381, −4.55834684131140031321784081525, −3.16174551478824360442184908445, −2.25610582463945759407900622794, 0,
2.25610582463945759407900622794, 3.16174551478824360442184908445, 4.55834684131140031321784081525, 5.65147980657085516848049895381, 6.33560598867710411281522365718, 7.53862789028003683931489821585, 8.146906664648363880391573042940, 9.559628331745101664415344236205, 9.954901377829594152119131032207
