| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s − 2·9-s − 11-s − 12-s + 5·13-s + 14-s + 16-s − 3·17-s + 2·18-s + 4·19-s + 21-s + 22-s − 4·23-s + 24-s − 5·26-s + 5·27-s − 28-s − 2·29-s + 2·31-s − 32-s + 33-s + 3·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s − 0.353·8-s − 2/3·9-s − 0.301·11-s − 0.288·12-s + 1.38·13-s + 0.267·14-s + 1/4·16-s − 0.727·17-s + 0.471·18-s + 0.917·19-s + 0.218·21-s + 0.213·22-s − 0.834·23-s + 0.204·24-s − 0.980·26-s + 0.962·27-s − 0.188·28-s − 0.371·29-s + 0.359·31-s − 0.176·32-s + 0.174·33-s + 0.514·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.397000768\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.397000768\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 293 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 7 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−13.70452762406138, −13.25258053693340, −12.71632182856104, −12.03445921500480, −11.68323036814263, −11.23423136163887, −10.82137461072072, −10.32770244587429, −9.851248330274526, −9.195334640415403, −8.759056656783536, −8.401868767070205, −7.728033721721602, −7.274841104966716, −6.558492809099138, −6.164756944323392, −5.690864958611930, −5.300042494278041, −4.358141682340387, −3.835610760236811, −3.158336332746871, −2.534940714617023, −1.923855580368334, −0.8929923083525991, −0.5664560374622428,
0.5664560374622428, 0.8929923083525991, 1.923855580368334, 2.534940714617023, 3.158336332746871, 3.835610760236811, 4.358141682340387, 5.300042494278041, 5.690864958611930, 6.164756944323392, 6.558492809099138, 7.274841104966716, 7.728033721721602, 8.401868767070205, 8.759056656783536, 9.195334640415403, 9.851248330274526, 10.32770244587429, 10.82137461072072, 11.23423136163887, 11.68323036814263, 12.03445921500480, 12.71632182856104, 13.25258053693340, 13.70452762406138
