| L(s) = 1 | + (−0.501 − 1.32i)2-s + (−2.51 + 1.34i)3-s + (−1.49 + 1.32i)4-s + (0.634 + 0.773i)5-s + (3.04 + 2.65i)6-s + (1.35 − 0.902i)7-s + (2.50 + 1.31i)8-s + (2.86 − 4.28i)9-s + (0.704 − 1.22i)10-s + (−1.04 + 0.317i)11-s + (1.98 − 5.35i)12-s + (−2.64 − 2.16i)13-s + (−1.86 − 1.33i)14-s + (−2.63 − 1.09i)15-s + (0.484 − 3.97i)16-s + (5.73 − 2.37i)17-s + ⋯ |
| L(s) = 1 | + (−0.354 − 0.935i)2-s + (−1.45 + 0.777i)3-s + (−0.748 + 0.662i)4-s + (0.283 + 0.345i)5-s + (1.24 + 1.08i)6-s + (0.510 − 0.340i)7-s + (0.885 + 0.465i)8-s + (0.954 − 1.42i)9-s + (0.222 − 0.387i)10-s + (−0.316 + 0.0958i)11-s + (0.573 − 1.54i)12-s + (−0.733 − 0.601i)13-s + (−0.499 − 0.356i)14-s + (−0.681 − 0.282i)15-s + (0.121 − 0.992i)16-s + (1.39 − 0.576i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.442 - 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.442 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.477603 + 0.297028i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.477603 + 0.297028i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.501 + 1.32i)T \) |
| 5 | \( 1 + (-0.634 - 0.773i)T \) |
| good | 3 | \( 1 + (2.51 - 1.34i)T + (1.66 - 2.49i)T^{2} \) |
| 7 | \( 1 + (-1.35 + 0.902i)T + (2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (1.04 - 0.317i)T + (9.14 - 6.11i)T^{2} \) |
| 13 | \( 1 + (2.64 + 2.16i)T + (2.53 + 12.7i)T^{2} \) |
| 17 | \( 1 + (-5.73 + 2.37i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (-0.629 - 6.39i)T + (-18.6 + 3.70i)T^{2} \) |
| 23 | \( 1 + (4.91 - 0.977i)T + (21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (1.03 - 3.40i)T + (-24.1 - 16.1i)T^{2} \) |
| 31 | \( 1 + (-5.21 - 5.21i)T + 31iT^{2} \) |
| 37 | \( 1 + (-2.05 - 0.202i)T + (36.2 + 7.21i)T^{2} \) |
| 41 | \( 1 + (-0.612 - 3.07i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (6.01 + 3.21i)T + (23.8 + 35.7i)T^{2} \) |
| 47 | \( 1 + (-2.39 - 5.77i)T + (-33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (-1.46 - 4.83i)T + (-44.0 + 29.4i)T^{2} \) |
| 59 | \( 1 + (2.88 - 2.36i)T + (11.5 - 57.8i)T^{2} \) |
| 61 | \( 1 + (-2.41 - 4.52i)T + (-33.8 + 50.7i)T^{2} \) |
| 67 | \( 1 + (-4.86 - 9.09i)T + (-37.2 + 55.7i)T^{2} \) |
| 71 | \( 1 + (4.22 + 6.32i)T + (-27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (6.47 + 4.32i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (6.20 - 14.9i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (8.72 - 0.858i)T + (81.4 - 16.1i)T^{2} \) |
| 89 | \( 1 + (-9.45 - 1.88i)T + (82.2 + 34.0i)T^{2} \) |
| 97 | \( 1 + (11.6 + 11.6i)T + 97iT^{2} \) |
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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
−10.47084451131584621826865770342, −10.21306270754317393678027818340, −9.633392418953004440187945114254, −8.131386600839839816274269539767, −7.34110390861299883781884544293, −5.83517631571839540081062930619, −5.16383768745143211826704866675, −4.23381690503139131207725116647, −3.06390911262373521395278297294, −1.25251778869989535484832086682,
0.48014495564342225564360706825, 1.88946791730774860295308929577, 4.50789016335655214245370638795, 5.27854549253606959020372678985, 5.93238368901123028387168491255, 6.75532313515282802657591550074, 7.63847216793719427178150718229, 8.370806332718049112390596809065, 9.624298777481926600100833465025, 10.29290276651839142476745879023
