L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.500 + 0.866i)4-s + (0.5 + 0.866i)5-s − 3·8-s − 0.999·10-s + (−2 + 3.46i)11-s + (1 + 1.73i)13-s + (0.500 − 0.866i)16-s − 2·17-s + 4·19-s + (−0.499 + 0.866i)20-s + (−1.99 − 3.46i)22-s + (−0.499 + 0.866i)25-s − 1.99·26-s + (−1 + 1.73i)29-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.250 + 0.433i)4-s + (0.223 + 0.387i)5-s − 1.06·8-s − 0.316·10-s + (−0.603 + 1.04i)11-s + (0.277 + 0.480i)13-s + (0.125 − 0.216i)16-s − 0.485·17-s + 0.917·19-s + (−0.111 + 0.193i)20-s + (−0.426 − 0.738i)22-s + (−0.0999 + 0.173i)25-s − 0.392·26-s + (−0.185 + 0.321i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.363964 + 0.999984i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.363964 + 0.999984i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
good | 2 | \( 1 + (0.5 - 0.866i)T + (-1 - 1.73i)T^{2} \) |
| 7 | \( 1 + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2 - 3.46i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1 - 1.73i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1 - 1.73i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 10T + 37T^{2} \) |
| 41 | \( 1 + (-5 - 8.66i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2 - 3.46i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4 + 6.92i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 10T + 53T^{2} \) |
| 59 | \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6 + 10.3i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-6 + 10.3i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.63631279952971224303853714579, −10.64152928560301242275577443534, −9.625966598251987688812605038515, −8.810133370240374653294235948282, −7.70207052092179351489798616263, −7.08021719328771249134578233951, −6.19478176171239042880879059654, −4.94890624958160086104752971198, −3.47803764713987674617217746521, −2.20696536058656135890082817817,
0.76517041338047048148171756520, 2.32517192390184871962881805736, 3.54463189125455891335815274182, 5.30528964311774442667302699047, 5.88842810105885743785816001124, 7.18235048403042326045897491627, 8.474299849568466977957851395955, 9.121950071644673854643705551091, 10.20107499208583961364476901053, 10.77963890822316167601103086561