| L(s) = 1 | + (−2.44 − 1.08i)2-s + (−0.216 + 2.06i)3-s + (3.44 + 3.83i)4-s − 2.87·5-s + (2.77 − 4.80i)6-s + (2.21 + 2.46i)7-s + (−2.60 − 8.02i)8-s + (−1.27 − 0.270i)9-s + (7.01 + 3.12i)10-s + (3.29 − 0.699i)11-s + (−8.64 + 6.28i)12-s + (3.57 + 0.474i)13-s + (−2.74 − 8.43i)14-s + (0.622 − 5.92i)15-s + (−1.28 + 12.1i)16-s + (5.58 + 1.18i)17-s + ⋯ |
| L(s) = 1 | + (−1.72 − 0.769i)2-s + (−0.125 + 1.19i)3-s + (1.72 + 1.91i)4-s − 1.28·5-s + (1.13 − 1.96i)6-s + (0.838 + 0.931i)7-s + (−0.921 − 2.83i)8-s + (−0.424 − 0.0901i)9-s + (2.21 + 0.987i)10-s + (0.992 − 0.210i)11-s + (−2.49 + 1.81i)12-s + (0.991 + 0.131i)13-s + (−0.732 − 2.25i)14-s + (0.160 − 1.52i)15-s + (−0.320 + 3.04i)16-s + (1.35 + 0.287i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0900 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0900 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.351531 + 0.384767i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.351531 + 0.384767i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (-3.57 - 0.474i)T \) |
| 31 | \( 1 + (5.49 + 0.917i)T \) |
| good | 2 | \( 1 + (2.44 + 1.08i)T + (1.33 + 1.48i)T^{2} \) |
| 3 | \( 1 + (0.216 - 2.06i)T + (-2.93 - 0.623i)T^{2} \) |
| 5 | \( 1 + 2.87T + 5T^{2} \) |
| 7 | \( 1 + (-2.21 - 2.46i)T + (-0.731 + 6.96i)T^{2} \) |
| 11 | \( 1 + (-3.29 + 0.699i)T + (10.0 - 4.47i)T^{2} \) |
| 17 | \( 1 + (-5.58 - 1.18i)T + (15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 + (0.0720 + 0.685i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (5.71 - 6.35i)T + (-2.40 - 22.8i)T^{2} \) |
| 29 | \( 1 + (0.973 + 0.433i)T + (19.4 + 21.5i)T^{2} \) |
| 37 | \( 1 + (3.42 + 5.93i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-6.37 - 2.83i)T + (27.4 + 30.4i)T^{2} \) |
| 43 | \( 1 + (-0.641 - 6.10i)T + (-42.0 + 8.94i)T^{2} \) |
| 47 | \( 1 + (-4.86 - 3.53i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-0.595 - 1.83i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (9.89 - 4.40i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (-5.35 + 9.26i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.34 + 2.32i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (1.08 + 0.231i)T + (64.8 + 28.8i)T^{2} \) |
| 73 | \( 1 + (0.0340 - 0.104i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (1.57 + 4.85i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-0.383 + 0.278i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (12.1 - 2.57i)T + (81.3 - 36.1i)T^{2} \) |
| 97 | \( 1 + (-10.2 - 11.3i)T + (-10.1 + 96.4i)T^{2} \) |
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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
−11.30026243207104846679544803969, −10.67492318184810108639448508960, −9.515053296563266457032198701872, −9.038238435842725451392048946274, −8.096582559294045050257708757097, −7.53363379198822521078984625845, −5.81768597048630821155258423448, −4.03076686186144525567595289589, −3.43911325165900969589355860387, −1.52779253018386396513116096089,
0.70287317060108908656383337642, 1.60531060035671893223039863884, 4.01749724103596950075673089815, 5.87878390930440987119453026765, 6.91778538776820317710059945613, 7.45322011054414676390141968721, 8.060055462935625480311479816205, 8.697512719364007579462161312982, 10.08035124035453306401662020051, 10.90479313995469815823835789181
