L(s) = 1 | + (1 + 1.73i)5-s + (−2.5 + 0.866i)7-s + (2 − 3.46i)11-s + (−1.5 + 2.59i)13-s + (3.5 + 6.06i)17-s + (−2.5 + 4.33i)19-s + (2 + 3.46i)23-s + (0.500 − 0.866i)25-s + (−0.5 − 0.866i)29-s − 3·31-s + (−4 − 3.46i)35-s + (−5.5 + 9.52i)37-s + (−4.5 + 7.79i)41-s + (−2.5 − 4.33i)43-s − 3·47-s + ⋯ |
L(s) = 1 | + (0.447 + 0.774i)5-s + (−0.944 + 0.327i)7-s + (0.603 − 1.04i)11-s + (−0.416 + 0.720i)13-s + (0.848 + 1.47i)17-s + (−0.573 + 0.993i)19-s + (0.417 + 0.722i)23-s + (0.100 − 0.173i)25-s + (−0.0928 − 0.160i)29-s − 0.538·31-s + (−0.676 − 0.585i)35-s + (−0.904 + 1.56i)37-s + (−0.702 + 1.21i)41-s + (−0.381 − 0.660i)43-s − 0.437·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0477 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0477 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.891917 + 0.935586i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.891917 + 0.935586i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.5 - 0.866i)T \) |
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 | \( 1 + (-3.5 - 6.06i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.5 - 4.33i)T + (-9.5 - 16.4i)T^{2} \) |
| 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 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 + (5.5 - 9.52i)T + (-18.5 - 32.0i)T^{2} \) |
| 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 | \( 1 + 3T + 47T^{2} \) |
| 53 | \( 1 + (-1.5 - 2.59i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 7T + 59T^{2} \) |
| 61 | \( 1 - 3T + 61T^{2} \) |
| 67 | \( 1 - 13T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + (3.5 + 6.06i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + 9T + 79T^{2} \) |
| 83 | \( 1 + (-0.5 - 0.866i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-7.5 + 12.9i)T + (-44.5 - 77.0i)T^{2} \) |
| 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
−10.31313144760867943305963275787, −9.928632641684825750663932613682, −8.879464896301950656724252178718, −8.103468648819423980249647960001, −6.73584013999702334254314109400, −6.32589036732423119887338415682, −5.47038022724303532255153842673, −3.81509550361383763978718273989, −3.14894096657691983778769241616, −1.72881084394999007448461093778,
0.66143563828744786385086072334, 2.32836138723172585613220795480, 3.57689741879840005906462568462, 4.83102811244788963470027481560, 5.47343146164123109187481022185, 6.87302740285976810536360452389, 7.23101791822365787235798964611, 8.643162652326933815121367626502, 9.418319934230174856377597976382, 9.871751115675098161112653304834