L(s) = 1 | + (−0.781 − 0.623i)2-s + (0.318 − 0.661i)3-s + (0.222 + 0.974i)4-s + (−2.09 − 0.794i)5-s + (−0.661 + 0.318i)6-s + (2.42 + 1.04i)7-s + (0.433 − 0.900i)8-s + (1.53 + 1.92i)9-s + (1.13 + 1.92i)10-s + (−3.47 + 4.35i)11-s + (0.715 + 0.163i)12-s + (2.13 + 1.70i)13-s + (−1.24 − 2.33i)14-s + (−1.19 + 1.12i)15-s + (−0.900 + 0.433i)16-s + (0.971 + 0.221i)17-s + ⋯ |
L(s) = 1 | + (−0.552 − 0.440i)2-s + (0.183 − 0.381i)3-s + (0.111 + 0.487i)4-s + (−0.934 − 0.355i)5-s + (−0.270 + 0.130i)6-s + (0.918 + 0.396i)7-s + (0.153 − 0.318i)8-s + (0.511 + 0.641i)9-s + (0.360 + 0.608i)10-s + (−1.04 + 1.31i)11-s + (0.206 + 0.0471i)12-s + (0.592 + 0.472i)13-s + (−0.332 − 0.623i)14-s + (−0.307 + 0.291i)15-s + (−0.225 + 0.108i)16-s + (0.235 + 0.0537i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.260i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 - 0.260i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.00995 + 0.134010i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.00995 + 0.134010i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.781 + 0.623i)T \) |
| 5 | \( 1 + (2.09 + 0.794i)T \) |
| 7 | \( 1 + (-2.42 - 1.04i)T \) |
good | 3 | \( 1 + (-0.318 + 0.661i)T + (-1.87 - 2.34i)T^{2} \) |
| 11 | \( 1 + (3.47 - 4.35i)T + (-2.44 - 10.7i)T^{2} \) |
| 13 | \( 1 + (-2.13 - 1.70i)T + (2.89 + 12.6i)T^{2} \) |
| 17 | \( 1 + (-0.971 - 0.221i)T + (15.3 + 7.37i)T^{2} \) |
| 19 | \( 1 + 3.53T + 19T^{2} \) |
| 23 | \( 1 + (-4.50 + 1.02i)T + (20.7 - 9.97i)T^{2} \) |
| 29 | \( 1 + (-0.0726 + 0.318i)T + (-26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 - 7.02T + 31T^{2} \) |
| 37 | \( 1 + (2.14 + 0.489i)T + (33.3 + 16.0i)T^{2} \) |
| 41 | \( 1 + (-2.19 - 1.05i)T + (25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (-5.13 - 10.6i)T + (-26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (6.06 + 4.83i)T + (10.4 + 45.8i)T^{2} \) |
| 53 | \( 1 + (3.19 - 0.729i)T + (47.7 - 22.9i)T^{2} \) |
| 59 | \( 1 + (0.581 - 0.279i)T + (36.7 - 46.1i)T^{2} \) |
| 61 | \( 1 + (0.124 - 0.544i)T + (-54.9 - 26.4i)T^{2} \) |
| 67 | \( 1 + 11.4iT - 67T^{2} \) |
| 71 | \( 1 + (-2.88 - 12.6i)T + (-63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (-5.91 + 4.71i)T + (16.2 - 71.1i)T^{2} \) |
| 79 | \( 1 + 13.7T + 79T^{2} \) |
| 83 | \( 1 + (-5.15 + 4.11i)T + (18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (4.20 + 5.27i)T + (-19.8 + 86.7i)T^{2} \) |
| 97 | \( 1 + 1.20iT - 97T^{2} \) |
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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
−10.99617684482828215360672086456, −10.26162537901017281451331172585, −9.072392436311522732286337919455, −8.104256164429676089653790313531, −7.79140286606932368408539722549, −6.77911437776419287788593651011, −4.94127792287267708390353705600, −4.35656748819957182545968190686, −2.61872488692869484145863960511, −1.48662002382580959512047605368,
0.814165875616495826284000808450, 3.02864511394565577165297605829, 4.12860537760902390505853701441, 5.26575605931801868554866860486, 6.48825912703159185466708597994, 7.51144592718712057507662363364, 8.267172001335563956769769168634, 8.801723870137962372312487589117, 10.22759232158853352052684259904, 10.80237446798376038770666914760