L(s) = 1 | + (−1.27 + 0.605i)2-s + (1.39 + 1.39i)3-s + (1.26 − 1.54i)4-s + (2.16 − 2.16i)5-s + (−2.62 − 0.935i)6-s − i·7-s + (−0.681 + 2.74i)8-s + 0.871i·9-s + (−1.45 + 4.07i)10-s + (−3.09 + 3.09i)11-s + (3.91 − 0.391i)12-s + (1.75 + 1.75i)13-s + (0.605 + 1.27i)14-s + 6.02·15-s + (−0.791 − 3.92i)16-s − 5.20·17-s + ⋯ |
L(s) = 1 | + (−0.903 + 0.428i)2-s + (0.803 + 0.803i)3-s + (0.633 − 0.773i)4-s + (0.968 − 0.968i)5-s + (−1.06 − 0.381i)6-s − 0.377i·7-s + (−0.240 + 0.970i)8-s + 0.290i·9-s + (−0.460 + 1.28i)10-s + (−0.933 + 0.933i)11-s + (1.13 − 0.112i)12-s + (0.486 + 0.486i)13-s + (0.161 + 0.341i)14-s + 1.55·15-s + (−0.197 − 0.980i)16-s − 1.26·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.829 - 0.557i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.829 - 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.928372 + 0.283052i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.928372 + 0.283052i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.27 - 0.605i)T \) |
| 7 | \( 1 + iT \) |
good | 3 | \( 1 + (-1.39 - 1.39i)T + 3iT^{2} \) |
| 5 | \( 1 + (-2.16 + 2.16i)T - 5iT^{2} \) |
| 11 | \( 1 + (3.09 - 3.09i)T - 11iT^{2} \) |
| 13 | \( 1 + (-1.75 - 1.75i)T + 13iT^{2} \) |
| 17 | \( 1 + 5.20T + 17T^{2} \) |
| 19 | \( 1 + (0.851 + 0.851i)T + 19iT^{2} \) |
| 23 | \( 1 - 6.15iT - 23T^{2} \) |
| 29 | \( 1 + (6.24 + 6.24i)T + 29iT^{2} \) |
| 31 | \( 1 + 2.78T + 31T^{2} \) |
| 37 | \( 1 + (-4.11 + 4.11i)T - 37iT^{2} \) |
| 41 | \( 1 - 6.32iT - 41T^{2} \) |
| 43 | \( 1 + (-3.05 + 3.05i)T - 43iT^{2} \) |
| 47 | \( 1 - 3.60T + 47T^{2} \) |
| 53 | \( 1 + (-5.28 + 5.28i)T - 53iT^{2} \) |
| 59 | \( 1 + (7.13 - 7.13i)T - 59iT^{2} \) |
| 61 | \( 1 + (-1.03 - 1.03i)T + 61iT^{2} \) |
| 67 | \( 1 + (0.966 + 0.966i)T + 67iT^{2} \) |
| 71 | \( 1 + 10.0iT - 71T^{2} \) |
| 73 | \( 1 - 15.1iT - 73T^{2} \) |
| 79 | \( 1 + 6.61T + 79T^{2} \) |
| 83 | \( 1 + (7.41 + 7.41i)T + 83iT^{2} \) |
| 89 | \( 1 - 3.26iT - 89T^{2} \) |
| 97 | \( 1 + 7.66T + 97T^{2} \) |
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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
−13.81999090686044216263730349489, −13.01649011729891511947310364413, −11.23736040617645082297903704101, −10.00739035157310388853402854782, −9.396464650514175196968139841677, −8.696759617365539933064223524408, −7.39199563125755620943118252369, −5.81155618748778068886143249812, −4.45887692388295544178784924984, −2.09055935893573580568223504325,
2.15460867979968697916247934910, 2.99904967314338074963587456243, 6.03160710142085535713867559366, 7.12580118620505610700957926165, 8.282837338989729770745466932504, 9.055237527701256912654368629981, 10.54462886067023117615568408774, 10.97945754217153553912700322712, 12.74920259100619369303801641910, 13.34121045635478565039560317398