L(s) = 1 | + (−1 − 1.73i)5-s + (1 − 1.73i)7-s + (−2.5 + 4.33i)11-s + (−2 − 3.46i)13-s − 17-s + 5·19-s + (−2 − 3.46i)23-s + (0.500 − 0.866i)25-s + (−3 + 5.19i)29-s − 3.99·35-s − 10·37-s + (−1.5 − 2.59i)41-s + (−4.5 + 7.79i)43-s + (4 − 6.92i)47-s + (1.50 + 2.59i)49-s + ⋯ |
L(s) = 1 | + (−0.447 − 0.774i)5-s + (0.377 − 0.654i)7-s + (−0.753 + 1.30i)11-s + (−0.554 − 0.960i)13-s − 0.242·17-s + 1.14·19-s + (−0.417 − 0.722i)23-s + (0.100 − 0.173i)25-s + (−0.557 + 0.964i)29-s − 0.676·35-s − 1.64·37-s + (−0.234 − 0.405i)41-s + (−0.686 + 1.18i)43-s + (0.583 − 1.01i)47-s + (0.214 + 0.371i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-1 + 1.73i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.5 - 4.33i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2 + 3.46i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + T + 17T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 10T + 37T^{2} \) |
| 41 | \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.5 - 7.79i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4 + 6.92i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 12T + 53T^{2} \) |
| 59 | \( 1 + (3.5 + 6.06i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2 - 3.46i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.5 + 6.06i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + 13T + 73T^{2} \) |
| 79 | \( 1 + (1 - 1.73i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (6 - 10.3i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + (6.5 - 11.2i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.102305614975890504191282414187, −7.35471937385486807001580692978, −7.03074209533279978590071051711, −5.60012150798417781037204263355, −4.96995315083339340717391235074, −4.46863807552114620593511311250, −3.45959970081733602511193966304, −2.37676927484482147123076594496, −1.22978028588612698307556585371, 0,
1.72037730736037071303668463282, 2.77521285964084499544758669971, 3.41548886638407051892639874617, 4.40521692894745410535710831185, 5.52064339902540727923142245308, 5.80950411115642470215733894285, 7.13073154173498712315106021995, 7.34352531557538493067237072928, 8.395765242350164182762291041109