| L(s) = 1 | + 2·2-s + 2·4-s − 5-s − 2·10-s − 2·11-s − 4·16-s + 6·17-s − 3·19-s − 2·20-s − 4·22-s − 3·23-s − 4·25-s − 3·29-s − 3·31-s − 8·32-s + 12·34-s − 6·37-s − 6·38-s − 10·41-s − 43-s − 4·44-s − 6·46-s − 11·47-s − 8·50-s + 9·53-s + 2·55-s − 6·58-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 0.447·5-s − 0.632·10-s − 0.603·11-s − 16-s + 1.45·17-s − 0.688·19-s − 0.447·20-s − 0.852·22-s − 0.625·23-s − 4/5·25-s − 0.557·29-s − 0.538·31-s − 1.41·32-s + 2.05·34-s − 0.986·37-s − 0.973·38-s − 1.56·41-s − 0.152·43-s − 0.603·44-s − 0.884·46-s − 1.60·47-s − 1.13·50-s + 1.23·53-s + 0.269·55-s − 0.787·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.944910752\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.944910752\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−14.11679732043508, −13.50364553360484, −13.01923284750899, −12.81010668924441, −11.99805240319398, −11.83584616522539, −11.38496983460283, −10.66688517277722, −10.01122510652686, −9.834275894646588, −8.845470156183339, −8.358558480722934, −7.933241325276094, −7.097949245018708, −6.911676705113185, −5.994624651447966, −5.564197009215585, −5.256272119946807, −4.516775302380027, −3.960036517967738, −3.465309404873784, −3.054851213352457, −2.169496834505098, −1.635278672168322, −0.3365215039515283,
0.3365215039515283, 1.635278672168322, 2.169496834505098, 3.054851213352457, 3.465309404873784, 3.960036517967738, 4.516775302380027, 5.256272119946807, 5.564197009215585, 5.994624651447966, 6.911676705113185, 7.097949245018708, 7.933241325276094, 8.358558480722934, 8.845470156183339, 9.834275894646588, 10.01122510652686, 10.66688517277722, 11.38496983460283, 11.83584616522539, 11.99805240319398, 12.81010668924441, 13.01923284750899, 13.50364553360484, 14.11679732043508
