L(s) = 1 | + (1.5 + 0.866i)3-s + (−1 − 1.73i)5-s + (1.5 + 2.59i)9-s + (−2 + 3.46i)11-s + (−1.5 − 2.59i)13-s − 3.46i·15-s + 7·17-s + 5·19-s + (−2 − 3.46i)23-s + (0.500 − 0.866i)25-s + 5.19i·27-s + (0.5 − 0.866i)29-s + (1.5 + 2.59i)31-s + (−6 + 3.46i)33-s + 11·37-s + ⋯ |
L(s) = 1 | + (0.866 + 0.499i)3-s + (−0.447 − 0.774i)5-s + (0.5 + 0.866i)9-s + (−0.603 + 1.04i)11-s + (−0.416 − 0.720i)13-s − 0.894i·15-s + 1.69·17-s + 1.14·19-s + (−0.417 − 0.722i)23-s + (0.100 − 0.173i)25-s + 0.999i·27-s + (0.0928 − 0.160i)29-s + (0.269 + 0.466i)31-s + (−1.04 + 0.603i)33-s + 1.80·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.248783424\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.248783424\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.5 - 0.866i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (2 - 3.46i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.5 + 2.59i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 7T + 17T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.5 + 0.866i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.5 - 2.59i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 11T + 37T^{2} \) |
| 41 | \( 1 + (-4.5 - 7.79i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.5 - 4.33i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1.5 - 2.59i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 3T + 53T^{2} \) |
| 59 | \( 1 + (-3.5 - 6.06i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1.5 - 2.59i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.5 + 11.2i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 7T + 73T^{2} \) |
| 79 | \( 1 + (-4.5 + 7.79i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (0.5 - 0.866i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 15T + 89T^{2} \) |
| 97 | \( 1 + (-8.5 + 14.7i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.494369520059240105349093439046, −8.439381825879307724344873397904, −7.69929798365836044307133322510, −7.56525048000470160776068205357, −5.96383061153251407913677904124, −4.89091267378441973098772403982, −4.55742412940866144273103127435, −3.32918717017753630730606040679, −2.56852784842004776780560795607, −1.09449135929903545031904738485,
0.983440521401001918670678879355, 2.43113924164211559633267557438, 3.26810381607533865412834763725, 3.82924845079674152275958423076, 5.30774630498182352102861261643, 6.13151671014564510412031210945, 7.21508148579139680569296874029, 7.62200855230161455202228512280, 8.247968132402511121733142599034, 9.289373618114066976767908533012