L(s) = 1 | + 1.49i·2-s + (−0.707 + 0.707i)3-s − 0.225·4-s + (0.731 − 0.731i)5-s + (−1.05 − 1.05i)6-s + (−0.0312 − 0.0312i)7-s + 2.64i·8-s − 1.00i·9-s + (1.09 + 1.09i)10-s + (2.62 + 2.62i)11-s + (0.159 − 0.159i)12-s − 13-s + (0.0465 − 0.0465i)14-s + 1.03i·15-s − 4.39·16-s + (2.32 − 3.40i)17-s + ⋯ |
L(s) = 1 | + 1.05i·2-s + (−0.408 + 0.408i)3-s − 0.112·4-s + (0.327 − 0.327i)5-s + (−0.430 − 0.430i)6-s + (−0.0118 − 0.0118i)7-s + 0.936i·8-s − 0.333i·9-s + (0.345 + 0.345i)10-s + (0.792 + 0.792i)11-s + (0.0459 − 0.0459i)12-s − 0.277·13-s + (0.0124 − 0.0124i)14-s + 0.267i·15-s − 1.09·16-s + (0.565 − 0.825i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 663 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 - 0.661i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 663 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.530722 + 1.40368i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.530722 + 1.40368i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.707 - 0.707i)T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + (-2.32 + 3.40i)T \) |
good | 2 | \( 1 - 1.49iT - 2T^{2} \) |
| 5 | \( 1 + (-0.731 + 0.731i)T - 5iT^{2} \) |
| 7 | \( 1 + (0.0312 + 0.0312i)T + 7iT^{2} \) |
| 11 | \( 1 + (-2.62 - 2.62i)T + 11iT^{2} \) |
| 19 | \( 1 - 4.41iT - 19T^{2} \) |
| 23 | \( 1 + (-2.82 - 2.82i)T + 23iT^{2} \) |
| 29 | \( 1 + (1.50 - 1.50i)T - 29iT^{2} \) |
| 31 | \( 1 + (-0.427 + 0.427i)T - 31iT^{2} \) |
| 37 | \( 1 + (7.53 - 7.53i)T - 37iT^{2} \) |
| 41 | \( 1 + (-1.87 - 1.87i)T + 41iT^{2} \) |
| 43 | \( 1 + 3.13iT - 43T^{2} \) |
| 47 | \( 1 - 1.35T + 47T^{2} \) |
| 53 | \( 1 + 2.47iT - 53T^{2} \) |
| 59 | \( 1 + 6.41iT - 59T^{2} \) |
| 61 | \( 1 + (7.89 + 7.89i)T + 61iT^{2} \) |
| 67 | \( 1 + 0.572T + 67T^{2} \) |
| 71 | \( 1 + (-5.12 + 5.12i)T - 71iT^{2} \) |
| 73 | \( 1 + (-1.11 + 1.11i)T - 73iT^{2} \) |
| 79 | \( 1 + (-11.9 - 11.9i)T + 79iT^{2} \) |
| 83 | \( 1 + 5.05iT - 83T^{2} \) |
| 89 | \( 1 - 17.5T + 89T^{2} \) |
| 97 | \( 1 + (8.89 - 8.89i)T - 97iT^{2} \) |
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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
−10.81330662974776225360606532313, −9.745455394372043777193308056158, −9.192456174368227210305713221761, −8.063921878053111962078228649205, −7.16456003956765738924711885187, −6.47878296823147438814232322214, −5.39503210796783902581342971466, −4.91261076951850650906724823744, −3.47296722220154361447396974232, −1.71923708907459251071549680258,
0.891325117775065735013591077521, 2.18761481008388289920941809275, 3.22559597377655606154335475957, 4.37073024216121039773512768775, 5.82269356580946288722353106198, 6.55490599939288702541674078245, 7.39605864828036369718391826546, 8.708571473578721284784435623618, 9.526594919865017270262926971754, 10.69894519820782493576930498216