| L(s) = 1 | + (1.11 + 0.872i)2-s + (1.16 + 2.29i)3-s + (0.476 + 1.94i)4-s + (−1.22 − 1.86i)5-s + (−0.700 + 3.56i)6-s + (−0.707 − 0.707i)7-s + (−1.16 + 2.57i)8-s + (−2.12 + 2.91i)9-s + (0.265 − 3.15i)10-s + (1.52 + 2.09i)11-s + (−3.89 + 3.35i)12-s + (1.01 + 6.43i)13-s + (−0.169 − 1.40i)14-s + (2.84 − 4.99i)15-s + (−3.54 + 1.85i)16-s + (−6.85 − 3.49i)17-s + ⋯ |
| L(s) = 1 | + (0.786 + 0.617i)2-s + (0.673 + 1.32i)3-s + (0.238 + 0.971i)4-s + (−0.549 − 0.835i)5-s + (−0.286 + 1.45i)6-s + (−0.267 − 0.267i)7-s + (−0.412 + 0.911i)8-s + (−0.706 + 0.972i)9-s + (0.0838 − 0.996i)10-s + (0.458 + 0.631i)11-s + (−1.12 + 0.969i)12-s + (0.282 + 1.78i)13-s + (−0.0453 − 0.375i)14-s + (0.735 − 1.28i)15-s + (−0.886 + 0.462i)16-s + (−1.66 − 0.847i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.871 - 0.491i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.871 - 0.491i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.615492 + 2.34407i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.615492 + 2.34407i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.11 - 0.872i)T \) |
| 5 | \( 1 + (1.22 + 1.86i)T \) |
| 7 | \( 1 + (0.707 + 0.707i)T \) |
| good | 3 | \( 1 + (-1.16 - 2.29i)T + (-1.76 + 2.42i)T^{2} \) |
| 11 | \( 1 + (-1.52 - 2.09i)T + (-3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-1.01 - 6.43i)T + (-12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (6.85 + 3.49i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (-1.13 - 3.48i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-0.788 + 4.97i)T + (-21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (-3.64 - 1.18i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-3.56 + 1.15i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-8.69 + 1.37i)T + (35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (-1.06 - 0.776i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (-3.97 + 3.97i)T - 43iT^{2} \) |
| 47 | \( 1 + (-6.64 + 3.38i)T + (27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (-9.62 + 4.90i)T + (31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (6.33 + 4.60i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (1.96 - 1.42i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-2.71 + 5.31i)T + (-39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (2.81 + 0.913i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (3.73 + 0.591i)T + (69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (1.32 - 4.08i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-6.96 - 3.54i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + (-0.173 - 0.239i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (1.19 + 2.33i)T + (-57.0 + 78.4i)T^{2} \) |
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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
−10.91066750597972483082978226206, −9.583540200408995292218619124152, −9.015507706573605874318809248619, −8.438901854049980605148244439902, −7.21347626019231727038009144417, −6.39964279149441300188827936485, −4.87455544512166947666594994204, −4.28619488042944302620576561635, −3.94025038513813920343864282188, −2.44001875417033798264466161719,
0.947348839150548651659297010018, 2.58008534703973960560524603320, 3.01618676765207634822990306753, 4.20484866467652565360854651658, 5.86425497695230901857216818446, 6.45328469265923721387896057146, 7.34963207924555103208712135605, 8.234864069598426243984699559188, 9.144731629420612018314776798547, 10.46169384959211296808732544619
