L(s) = 1 | + (−1.34 − 0.421i)2-s + (−1.90 − 1.90i)3-s + (1.64 + 1.13i)4-s + (−0.987 − 2.00i)5-s + (1.77 + 3.38i)6-s + (−3.15 + 3.15i)7-s + (−1.73 − 2.23i)8-s + 4.28i·9-s + (0.485 + 3.12i)10-s − 1.76i·11-s + (−0.964 − 5.31i)12-s + (0.707 − 0.707i)13-s + (5.58 − 2.92i)14-s + (−1.94 + 5.71i)15-s + (1.40 + 3.74i)16-s + (2.56 + 2.56i)17-s + ⋯ |
L(s) = 1 | + (−0.954 − 0.298i)2-s + (−1.10 − 1.10i)3-s + (0.822 + 0.569i)4-s + (−0.441 − 0.897i)5-s + (0.723 + 1.38i)6-s + (−1.19 + 1.19i)7-s + (−0.614 − 0.788i)8-s + 1.42i·9-s + (0.153 + 0.988i)10-s − 0.533i·11-s + (−0.278 − 1.53i)12-s + (0.196 − 0.196i)13-s + (1.49 − 0.781i)14-s + (−0.502 + 1.47i)15-s + (0.351 + 0.936i)16-s + (0.622 + 0.622i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.371 - 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.371 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.111067 + 0.0752015i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.111067 + 0.0752015i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.34 + 0.421i)T \) |
| 5 | \( 1 + (0.987 + 2.00i)T \) |
| 13 | \( 1 + (-0.707 + 0.707i)T \) |
good | 3 | \( 1 + (1.90 + 1.90i)T + 3iT^{2} \) |
| 7 | \( 1 + (3.15 - 3.15i)T - 7iT^{2} \) |
| 11 | \( 1 + 1.76iT - 11T^{2} \) |
| 17 | \( 1 + (-2.56 - 2.56i)T + 17iT^{2} \) |
| 19 | \( 1 + 1.23T + 19T^{2} \) |
| 23 | \( 1 + (-2.73 - 2.73i)T + 23iT^{2} \) |
| 29 | \( 1 - 2.75iT - 29T^{2} \) |
| 31 | \( 1 - 4.10iT - 31T^{2} \) |
| 37 | \( 1 + (-1.55 - 1.55i)T + 37iT^{2} \) |
| 41 | \( 1 + 11.6T + 41T^{2} \) |
| 43 | \( 1 + (0.431 + 0.431i)T + 43iT^{2} \) |
| 47 | \( 1 + (-8.54 + 8.54i)T - 47iT^{2} \) |
| 53 | \( 1 + (10.2 - 10.2i)T - 53iT^{2} \) |
| 59 | \( 1 + 7.03T + 59T^{2} \) |
| 61 | \( 1 + 8.24T + 61T^{2} \) |
| 67 | \( 1 + (1.41 - 1.41i)T - 67iT^{2} \) |
| 71 | \( 1 - 11.4iT - 71T^{2} \) |
| 73 | \( 1 + (5.51 - 5.51i)T - 73iT^{2} \) |
| 79 | \( 1 + 6.65T + 79T^{2} \) |
| 83 | \( 1 + (3.92 + 3.92i)T + 83iT^{2} \) |
| 89 | \( 1 + 8.02iT - 89T^{2} \) |
| 97 | \( 1 + (-1.20 - 1.20i)T + 97iT^{2} \) |
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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
−12.16084522033474370750034277055, −11.47541608600217258429876347038, −10.35663582368460174837555370081, −9.106233022764092715205310178094, −8.417736418805090909862156197642, −7.26625618839722920506840874016, −6.23568891935358668672722984334, −5.50904723096633959441106975004, −3.22653165728225866435590829352, −1.40048552394349729366520418402,
0.16929291466187613798202231580, 3.24391499845454324080243919342, 4.52034592691101558463536256638, 6.08217165606658549785169048517, 6.78158033577888641802506466673, 7.67468313336315939292273037376, 9.421311279025698287862977564580, 10.03038577062575787104740076756, 10.64114331462891779402868679266, 11.30176460671509131969074677631