L(s) = 1 | + (−1.71 + 0.211i)3-s + (−0.5 − 0.866i)5-s + (−0.719 + 1.24i)7-s + (2.91 − 0.728i)9-s + (0.675 − 1.17i)11-s + (2.76 + 4.78i)13-s + (1.04 + 1.38i)15-s + 4.82·17-s − 0.648·19-s + (0.972 − 2.29i)21-s + (4.45 + 7.71i)23-s + (−0.499 + 0.866i)25-s + (−4.84 + 1.86i)27-s + (3.58 − 6.21i)29-s + (2.32 + 4.02i)31-s + ⋯ |
L(s) = 1 | + (−0.992 + 0.122i)3-s + (−0.223 − 0.387i)5-s + (−0.271 + 0.470i)7-s + (0.970 − 0.242i)9-s + (0.203 − 0.353i)11-s + (0.766 + 1.32i)13-s + (0.269 + 0.357i)15-s + 1.16·17-s − 0.148·19-s + (0.212 − 0.500i)21-s + (0.928 + 1.60i)23-s + (−0.0999 + 0.173i)25-s + (−0.933 + 0.359i)27-s + (0.665 − 1.15i)29-s + (0.417 + 0.722i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.828 - 0.560i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.828 - 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.917960 + 0.281150i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.917960 + 0.281150i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.71 - 0.211i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
good | 7 | \( 1 + (0.719 - 1.24i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.675 + 1.17i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.76 - 4.78i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 4.82T + 17T^{2} \) |
| 19 | \( 1 + 0.648T + 19T^{2} \) |
| 23 | \( 1 + (-4.45 - 7.71i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.58 + 6.21i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.32 - 4.02i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 1.35T + 37T^{2} \) |
| 41 | \( 1 + (0.175 + 0.304i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.41 - 4.17i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.74 + 8.22i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 8.17T + 53T^{2} \) |
| 59 | \( 1 + (0.734 + 1.27i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.34 - 5.79i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.21 + 10.7i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 2.22T + 71T^{2} \) |
| 73 | \( 1 + 4.34T + 73T^{2} \) |
| 79 | \( 1 + (-6.52 + 11.3i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.63 + 4.56i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 11T + 89T^{2} \) |
| 97 | \( 1 + (8.79 - 15.2i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.73982777686304854318292351238, −10.77817229133538763254692844097, −9.655945338915570047149890288260, −8.950547184795784667720162007328, −7.67979553350881178386966274894, −6.52482726123532533916908684824, −5.74611534519411079612301020298, −4.67261369948393985556976389801, −3.50459121958554878090589360108, −1.32422863162694981497057729938,
0.919666056207958940391486868737, 3.11738144054325789498189335654, 4.42425081229198315781460175294, 5.59380555724354491844693425660, 6.54065320798195707825142845871, 7.38754880596196270674310065817, 8.411379316852050846781641840512, 9.912849401401543717366497020894, 10.55153268866612323406778350391, 11.17428501532692320108570215276