| L(s) = 1 | − 1.53·2-s + 1.34·4-s − 5-s − 0.532·8-s + 1.53·10-s − 0.532·16-s + 1.87·17-s − 1.87·19-s − 1.34·20-s − 0.347·23-s + 25-s + 1.53·31-s + 1.34·32-s − 2.87·34-s + 2.87·38-s + 0.532·40-s + 0.532·46-s + 47-s + 49-s − 1.53·50-s − 0.347·53-s + 1.53·61-s − 2.34·62-s − 1.53·64-s + 2.53·68-s − 2.53·76-s + 0.347·79-s + ⋯ |
| L(s) = 1 | − 1.53·2-s + 1.34·4-s − 5-s − 0.532·8-s + 1.53·10-s − 0.532·16-s + 1.87·17-s − 1.87·19-s − 1.34·20-s − 0.347·23-s + 25-s + 1.53·31-s + 1.34·32-s − 2.87·34-s + 2.87·38-s + 0.532·40-s + 0.532·46-s + 47-s + 49-s − 1.53·50-s − 0.347·53-s + 1.53·61-s − 2.34·62-s − 1.53·64-s + 2.53·68-s − 2.53·76-s + 0.347·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1215 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1215 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4069893075\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4069893075\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| good | 2 | \( 1 + 1.53T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - 1.87T + T^{2} \) |
| 19 | \( 1 + 1.87T + T^{2} \) |
| 23 | \( 1 + 0.347T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - 1.53T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T + T^{2} \) |
| 53 | \( 1 + 0.347T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 1.53T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - 0.347T + T^{2} \) |
| 83 | \( 1 - 1.87T + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.01560863708840977353991910405, −8.948028478210457781886713969945, −8.291867759144774257263739097355, −7.80423598441035057704704438528, −7.00484362942174626563346358042, −6.08287050773859682180577955196, −4.70103732842276672144431658969, −3.69978449927477500861679657673, −2.38671046857683219749119513085, −0.895066291246857968919851878791,
0.895066291246857968919851878791, 2.38671046857683219749119513085, 3.69978449927477500861679657673, 4.70103732842276672144431658969, 6.08287050773859682180577955196, 7.00484362942174626563346358042, 7.80423598441035057704704438528, 8.291867759144774257263739097355, 8.948028478210457781886713969945, 10.01560863708840977353991910405
