| L(s) = 1 | − 3-s − 7-s + 9-s + 4·11-s + 5·13-s + 8·17-s + 5·19-s + 21-s + 8·23-s − 27-s − 3·29-s + 31-s − 4·33-s − 8·37-s − 5·39-s − 11·41-s − 43-s + 10·47-s − 6·49-s − 8·51-s + 2·53-s − 5·57-s + 3·61-s − 63-s + 11·67-s − 8·69-s + 6·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s + 1.20·11-s + 1.38·13-s + 1.94·17-s + 1.14·19-s + 0.218·21-s + 1.66·23-s − 0.192·27-s − 0.557·29-s + 0.179·31-s − 0.696·33-s − 1.31·37-s − 0.800·39-s − 1.71·41-s − 0.152·43-s + 1.45·47-s − 6/7·49-s − 1.12·51-s + 0.274·53-s − 0.662·57-s + 0.384·61-s − 0.125·63-s + 1.34·67-s − 0.963·69-s + 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.138465121\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.138465121\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 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
−14.34889851680883, −13.93771364799077, −13.58584387640791, −12.80964942494143, −12.41701786192839, −11.91196588542960, −11.36945942677481, −11.07924468733307, −10.28259794924818, −9.893128424231078, −9.325717690181111, −8.802229237648221, −8.268238102284973, −7.496897893062796, −6.968388051790944, −6.550329159868140, −5.899151818719718, −5.331308703002716, −4.995323888997890, −3.912234744419817, −3.425100471444196, −3.219515428589701, −1.866519778174397, −1.124995053986244, −0.8059926595130398,
0.8059926595130398, 1.124995053986244, 1.866519778174397, 3.219515428589701, 3.425100471444196, 3.912234744419817, 4.995323888997890, 5.331308703002716, 5.899151818719718, 6.550329159868140, 6.968388051790944, 7.496897893062796, 8.268238102284973, 8.802229237648221, 9.325717690181111, 9.893128424231078, 10.28259794924818, 11.07924468733307, 11.36945942677481, 11.91196588542960, 12.41701786192839, 12.80964942494143, 13.58584387640791, 13.93771364799077, 14.34889851680883
