| L(s) = 1 | + (−1.41 − 0.0689i)2-s + (0.145 − 1.17i)3-s + (1.99 + 0.194i)4-s + (1.12 − 0.252i)5-s + (−0.285 + 1.65i)6-s + (0.487 − 0.361i)7-s + (−2.79 − 0.412i)8-s + (1.54 + 0.387i)9-s + (−1.60 + 0.279i)10-s + (4.01 + 0.295i)11-s + (0.517 − 2.31i)12-s + (−2.70 − 1.90i)13-s + (−0.714 + 0.477i)14-s + (−0.133 − 1.35i)15-s + (3.92 + 0.775i)16-s + (0.966 + 0.0952i)17-s + ⋯ |
| L(s) = 1 | + (−0.998 − 0.0487i)2-s + (0.0837 − 0.678i)3-s + (0.995 + 0.0974i)4-s + (0.502 − 0.112i)5-s + (−0.116 + 0.673i)6-s + (0.184 − 0.136i)7-s + (−0.989 − 0.145i)8-s + (0.516 + 0.129i)9-s + (−0.507 + 0.0882i)10-s + (1.20 + 0.0892i)11-s + (0.149 − 0.667i)12-s + (−0.749 − 0.528i)13-s + (−0.190 + 0.127i)14-s + (−0.0345 − 0.350i)15-s + (0.981 + 0.193i)16-s + (0.234 + 0.0230i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.664 + 0.747i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.664 + 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.07075 - 0.480838i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.07075 - 0.480838i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.41 + 0.0689i)T \) |
| good | 3 | \( 1 + (-0.145 + 1.17i)T + (-2.91 - 0.728i)T^{2} \) |
| 5 | \( 1 + (-1.12 + 0.252i)T + (4.51 - 2.13i)T^{2} \) |
| 7 | \( 1 + (-0.487 + 0.361i)T + (2.03 - 6.69i)T^{2} \) |
| 11 | \( 1 + (-4.01 - 0.295i)T + (10.8 + 1.61i)T^{2} \) |
| 13 | \( 1 + (2.70 + 1.90i)T + (4.37 + 12.2i)T^{2} \) |
| 17 | \( 1 + (-0.966 - 0.0952i)T + (16.6 + 3.31i)T^{2} \) |
| 19 | \( 1 + (-2.08 - 5.39i)T + (-14.0 + 12.7i)T^{2} \) |
| 23 | \( 1 + (-4.15 - 0.204i)T + (22.8 + 2.25i)T^{2} \) |
| 29 | \( 1 + (1.38 + 2.75i)T + (-17.2 + 23.2i)T^{2} \) |
| 31 | \( 1 + (0.0787 + 0.395i)T + (-28.6 + 11.8i)T^{2} \) |
| 37 | \( 1 + (2.71 + 2.58i)T + (1.81 + 36.9i)T^{2} \) |
| 41 | \( 1 + (2.66 + 1.26i)T + (26.0 + 31.6i)T^{2} \) |
| 43 | \( 1 + (-0.307 + 0.394i)T + (-10.4 - 41.7i)T^{2} \) |
| 47 | \( 1 + (-10.1 + 3.06i)T + (39.0 - 26.1i)T^{2} \) |
| 53 | \( 1 + (-6.39 + 12.6i)T + (-31.5 - 42.5i)T^{2} \) |
| 59 | \( 1 + (3.20 + 4.55i)T + (-19.8 + 55.5i)T^{2} \) |
| 61 | \( 1 + (0.808 + 0.458i)T + (31.3 + 52.3i)T^{2} \) |
| 67 | \( 1 + (-3.98 + 1.10i)T + (57.4 - 34.4i)T^{2} \) |
| 71 | \( 1 + (0.208 - 0.347i)T + (-33.4 - 62.6i)T^{2} \) |
| 73 | \( 1 + (6.19 + 4.59i)T + (21.1 + 69.8i)T^{2} \) |
| 79 | \( 1 + (2.94 + 5.51i)T + (-43.8 + 65.6i)T^{2} \) |
| 83 | \( 1 + (5.12 - 4.88i)T + (4.07 - 82.9i)T^{2} \) |
| 89 | \( 1 + (-7.40 + 0.363i)T + (88.5 - 8.72i)T^{2} \) |
| 97 | \( 1 + (8.11 - 12.1i)T + (-37.1 - 89.6i)T^{2} \) |
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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
−10.51625456668899762734035827644, −9.818767380821342303597351736373, −9.107525183863448698186252818752, −7.957915745021986054858646361451, −7.34231315565934575955283829711, −6.48705172678982845972361011080, −5.44506907878647165205961061844, −3.72651073027205494472071000612, −2.14071287001510607327546852844, −1.18000912962597207210957929820,
1.41504304763193910347044918347, 2.85789550062148393856918482870, 4.26973190856028254343402773262, 5.50697535982733768581422593334, 6.76896952119501998253415767378, 7.27498587305826318698716122861, 8.772679878523345358821623127490, 9.292800174083797700940674425189, 9.900993165086378354536830136413, 10.77868385257728419460537425017
