L(s) = 1 | + 3-s + (−2 + i)5-s + 2·7-s − 2·9-s + i·13-s + (−2 + i)15-s + (−1 + 4i)17-s + 5·19-s + 2·21-s − 4·23-s + (3 − 4i)25-s − 5·27-s + 9i·29-s + 5i·31-s + (−4 + 2i)35-s + ⋯ |
L(s) = 1 | + 0.577·3-s + (−0.894 + 0.447i)5-s + 0.755·7-s − 0.666·9-s + 0.277i·13-s + (−0.516 + 0.258i)15-s + (−0.242 + 0.970i)17-s + 1.14·19-s + 0.436·21-s − 0.834·23-s + (0.600 − 0.800i)25-s − 0.962·27-s + 1.67i·29-s + 0.898i·31-s + (−0.676 + 0.338i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.216 - 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.216 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.311130245\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.311130245\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (2 - i)T \) |
| 17 | \( 1 + (1 - 4i)T \) |
good | 3 | \( 1 - T + 3T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - iT - 13T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 9iT - 29T^{2} \) |
| 31 | \( 1 - 5iT - 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 10iT - 41T^{2} \) |
| 43 | \( 1 + 6iT - 43T^{2} \) |
| 47 | \( 1 - 7iT - 47T^{2} \) |
| 53 | \( 1 - iT - 53T^{2} \) |
| 59 | \( 1 - 5T + 59T^{2} \) |
| 61 | \( 1 + 5iT - 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 - 5iT - 71T^{2} \) |
| 73 | \( 1 + 11T + 73T^{2} \) |
| 79 | \( 1 - 16iT - 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 - 5T + 89T^{2} \) |
| 97 | \( 1 + 7T + 97T^{2} \) |
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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
−9.787760048323376705649460939080, −8.711920194574137199254694968870, −8.295165576441118961765660550889, −7.55928192609360918047615572455, −6.73811754420882217465280191838, −5.62885469716431337555275888599, −4.64067276973134209478790291242, −3.63617002174974162801158829612, −2.90489476939258268634211816956, −1.56839978927714775168895866166,
0.50026485977115467379362761482, 2.14435649821642831584975571915, 3.23926019641678786192941888764, 4.16454962193137831759630703757, 5.07740430291968729544691308177, 5.90357921789425646246172294165, 7.32868227586928008437788696652, 7.82232140084351518844120980245, 8.461431721359408246489103124804, 9.204499321350239169982599014922