| L(s) = 1 | + (−1.40 − 0.626i)2-s + (0.286 − 2.72i)3-s + (0.250 + 0.278i)4-s − 3.15·5-s + (−2.11 + 3.66i)6-s + (2.27 + 2.53i)7-s + (0.774 + 2.38i)8-s + (−4.42 − 0.940i)9-s + (4.44 + 1.97i)10-s + (−1.30 + 0.277i)11-s + (0.830 − 0.603i)12-s + (1.20 + 3.39i)13-s + (−1.62 − 4.99i)14-s + (−0.905 + 8.61i)15-s + (0.481 − 4.58i)16-s + (−6.68 − 1.42i)17-s + ⋯ |
| L(s) = 1 | + (−0.995 − 0.443i)2-s + (0.165 − 1.57i)3-s + (0.125 + 0.139i)4-s − 1.41·5-s + (−0.862 + 1.49i)6-s + (0.861 + 0.956i)7-s + (0.273 + 0.842i)8-s + (−1.47 − 0.313i)9-s + (1.40 + 0.625i)10-s + (−0.394 + 0.0837i)11-s + (0.239 − 0.174i)12-s + (0.334 + 0.942i)13-s + (−0.433 − 1.33i)14-s + (−0.233 + 2.22i)15-s + (0.120 − 1.14i)16-s + (−1.62 − 0.344i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 - 0.614i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.788 - 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.249829 + 0.0858769i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.249829 + 0.0858769i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (-1.20 - 3.39i)T \) |
| 31 | \( 1 + (5.55 + 0.370i)T \) |
| good | 2 | \( 1 + (1.40 + 0.626i)T + (1.33 + 1.48i)T^{2} \) |
| 3 | \( 1 + (-0.286 + 2.72i)T + (-2.93 - 0.623i)T^{2} \) |
| 5 | \( 1 + 3.15T + 5T^{2} \) |
| 7 | \( 1 + (-2.27 - 2.53i)T + (-0.731 + 6.96i)T^{2} \) |
| 11 | \( 1 + (1.30 - 0.277i)T + (10.0 - 4.47i)T^{2} \) |
| 17 | \( 1 + (6.68 + 1.42i)T + (15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 + (-0.594 - 5.65i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (0.381 - 0.423i)T + (-2.40 - 22.8i)T^{2} \) |
| 29 | \( 1 + (-3.89 - 1.73i)T + (19.4 + 21.5i)T^{2} \) |
| 37 | \( 1 + (5.67 + 9.82i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.33 - 1.92i)T + (27.4 + 30.4i)T^{2} \) |
| 43 | \( 1 + (-0.690 - 6.57i)T + (-42.0 + 8.94i)T^{2} \) |
| 47 | \( 1 + (-2.58 - 1.87i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-1.89 - 5.84i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (6.58 - 2.93i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (5.97 - 10.3i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.48 - 6.03i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (4.69 + 0.998i)T + (64.8 + 28.8i)T^{2} \) |
| 73 | \( 1 + (0.693 - 2.13i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (1.60 + 4.93i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (10.4 - 7.59i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-2.97 + 0.633i)T + (81.3 - 36.1i)T^{2} \) |
| 97 | \( 1 + (0.488 + 0.542i)T + (-10.1 + 96.4i)T^{2} \) |
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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
−11.47016540412886392871543768103, −10.77986210765370090590901348371, −9.022491342820219069708600646777, −8.599683340443544731949617619351, −7.79636800616382331685283972769, −7.20796750841838686633973017011, −5.81797482110728180575173983379, −4.39465056288174732405345782321, −2.46191772828841807840005431206, −1.51472019535233878272003406228,
0.24464950640277588661695080850, 3.40313209561184031090903431329, 4.26130641023452475672737049608, 4.87721466181192096969644602792, 6.88285275067219550199639067695, 7.82537354697717478236802972658, 8.464118602116344832256355238708, 9.120046742516105034876116851783, 10.42097852487390866816315664349, 10.74696341546648791367782290736
