L(s) = 1 | + (−0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (0.792 + 1.37i)5-s − 0.999·6-s + (−2.62 − 0.358i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (0.792 − 1.37i)10-s + (0.5 − 0.866i)11-s + (0.499 + 0.866i)12-s + 2.82·13-s + (1 + 2.44i)14-s + 1.58·15-s + (−0.5 − 0.866i)16-s + (2.12 − 3.67i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.354 + 0.614i)5-s − 0.408·6-s + (−0.990 − 0.135i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (0.250 − 0.434i)10-s + (0.150 − 0.261i)11-s + (0.144 + 0.249i)12-s + 0.784·13-s + (0.267 + 0.654i)14-s + 0.409·15-s + (−0.125 − 0.216i)16-s + (0.514 − 0.891i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.198 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.802749 - 0.981178i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.802749 - 0.981178i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (2.62 + 0.358i)T \) |
| 19 | \( 1 + (0.5 + 0.866i)T \) |
good | 5 | \( 1 + (-0.792 - 1.37i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 17 | \( 1 + (-2.12 + 3.67i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (1.70 + 2.95i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 2.17T + 29T^{2} \) |
| 31 | \( 1 + (-5.32 + 9.22i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.121 + 0.210i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 0.585T + 41T^{2} \) |
| 43 | \( 1 - 6.24T + 43T^{2} \) |
| 47 | \( 1 + (1.58 + 2.74i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.914 - 1.58i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3.03 - 5.25i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.41 - 4.18i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1 + 1.73i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 7.89T + 71T^{2} \) |
| 73 | \( 1 + (-7.41 + 12.8i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.0857 - 0.148i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 17.1T + 83T^{2} \) |
| 89 | \( 1 + (1.87 + 3.25i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 5.58T + 97T^{2} \) |
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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
−10.02918356727781457107830763543, −9.312539569981503718532147208786, −8.457487669278599402285477276248, −7.49667934836725759437598019541, −6.57089355671868198704372718759, −5.92300055147662346052161628481, −4.24211277218324257306097834592, −3.13088137543958099007457933335, −2.43503362928647196856039301507, −0.75892320862623553526495040214,
1.41296649158129671546084947271, 3.16270185061131601814143916962, 4.20097969927766368715433452917, 5.36796568687533943277449717096, 6.10600809800832053534029359640, 7.00271614096463559887752895961, 8.238328008119692624583709414550, 8.764135182241552881162159478741, 9.631213843771337371779962372232, 10.12300005515334745799390077874