L(s) = 1 | + (1 + 1.73i)4-s + (−0.5 + 2.59i)7-s + 2·13-s + (−1.99 + 3.46i)16-s + (3.5 − 6.06i)19-s + (2.5 + 4.33i)25-s + (−5 + 1.73i)28-s + (−5.5 − 9.52i)31-s + (5 − 8.66i)37-s − 13·43-s + (−6.5 − 2.59i)49-s + (2 + 3.46i)52-s + (6.5 − 11.2i)61-s − 7.99·64-s + (8 + 13.8i)67-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)4-s + (−0.188 + 0.981i)7-s + 0.554·13-s + (−0.499 + 0.866i)16-s + (0.802 − 1.39i)19-s + (0.5 + 0.866i)25-s + (−0.944 + 0.327i)28-s + (−0.987 − 1.71i)31-s + (0.821 − 1.42i)37-s − 1.98·43-s + (−0.928 − 0.371i)49-s + (0.277 + 0.480i)52-s + (0.832 − 1.44i)61-s − 0.999·64-s + (0.977 + 1.69i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.14441 + 0.567317i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.14441 + 0.567317i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (0.5 - 2.59i)T \) |
good | 2 | \( 1 + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.5 + 6.06i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + (5.5 + 9.52i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-5 + 8.66i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 13T + 43T^{2} \) |
| 47 | \( 1 + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.5 + 11.2i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-8 - 13.8i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (-3.5 - 6.06i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 5T + 97T^{2} \) |
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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
−12.80444859314965608835283090665, −11.56133513843806040725467735973, −11.19686013625458801930042760087, −9.513890445895080945335261376144, −8.710205457721235052209137861185, −7.60468031972361752969737633225, −6.56156358666153370563150050580, −5.31191683311091176991980917918, −3.62604658009176267743390914269, −2.39735138305506796527866923784,
1.37407463838619250781379380582, 3.40216835495597057193017243085, 4.95364039631166980593041056903, 6.21676046051896484871812868536, 7.09273544245850110200423430314, 8.322709809806686420757083655108, 9.795318746449690797632937535154, 10.38411073186735632279211533145, 11.28553459171877990901760917024, 12.34642365760301109895533903481