| L(s) = 1 | + (1.41 − 0.00887i)2-s + (0.755 − 0.654i)3-s + (1.99 − 0.0250i)4-s + (2.01 − 0.290i)5-s + (1.06 − 0.932i)6-s + (0.214 − 0.470i)7-s + (2.82 − 0.0532i)8-s + (0.142 − 0.989i)9-s + (2.85 − 0.428i)10-s + (−0.754 + 2.56i)11-s + (1.49 − 1.32i)12-s + (−5.11 + 2.33i)13-s + (0.299 − 0.666i)14-s + (1.33 − 1.54i)15-s + (3.99 − 0.100i)16-s + (−3.23 − 2.07i)17-s + ⋯ |
| L(s) = 1 | + (0.999 − 0.00627i)2-s + (0.436 − 0.378i)3-s + (0.999 − 0.0125i)4-s + (0.902 − 0.129i)5-s + (0.433 − 0.380i)6-s + (0.0811 − 0.177i)7-s + (0.999 − 0.0188i)8-s + (0.0474 − 0.329i)9-s + (0.902 − 0.135i)10-s + (−0.227 + 0.774i)11-s + (0.431 − 0.383i)12-s + (−1.41 + 0.647i)13-s + (0.0800 − 0.178i)14-s + (0.344 − 0.398i)15-s + (0.999 − 0.0250i)16-s + (−0.784 − 0.504i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 + 0.345i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.938 + 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.28759 - 0.585461i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.28759 - 0.585461i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.41 + 0.00887i)T \) |
| 3 | \( 1 + (-0.755 + 0.654i)T \) |
| 23 | \( 1 + (2.90 + 3.81i)T \) |
| good | 5 | \( 1 + (-2.01 + 0.290i)T + (4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (-0.214 + 0.470i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (0.754 - 2.56i)T + (-9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (5.11 - 2.33i)T + (8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (3.23 + 2.07i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (-0.853 - 1.32i)T + (-7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (1.80 - 2.80i)T + (-12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-2.62 + 3.02i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (6.67 + 0.960i)T + (35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (-0.979 - 6.81i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (-7.51 + 6.50i)T + (6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 - 7.60T + 47T^{2} \) |
| 53 | \( 1 + (1.29 + 0.592i)T + (34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (0.925 - 0.422i)T + (38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (4.46 + 3.87i)T + (8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (0.456 + 1.55i)T + (-56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (14.2 - 4.17i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (-10.3 + 6.62i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (3.77 + 8.27i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-5.36 - 0.770i)T + (79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (-8.92 - 10.3i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (0.0860 + 0.598i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−10.75902716665670361119577304342, −9.902887693537020564283273019776, −9.128658635714864916387267270520, −7.67520872681392541110019258375, −7.06235843959917799656193173228, −6.11768615970280814264754599924, −5.02498854539733163303654527919, −4.19634961324550978964413532041, −2.57275949159163970928752578568, −1.92630812082536602987584803168,
2.09447512200242956758568237016, 2.92865201171089455372141989446, 4.18032208579395675101357642394, 5.32846571710687683130440759759, 5.90260303208849302056275289712, 7.11577581154328273688010850864, 8.037533198663081529061032261901, 9.206576820714739563235374534861, 10.16322550983338068794002384789, 10.74168713188891433419322379529
