L(s) = 1 | + (−0.707 + 0.707i)3-s + (2.09 − 0.768i)5-s + (0.0117 − 0.0117i)7-s − 1.00i·9-s − 1.72i·11-s + (−2.37 + 2.37i)13-s + (−0.941 + 2.02i)15-s + (−1.66 + 1.66i)17-s + 6.73·19-s + 0.0165i·21-s + (4.79 + 0.0541i)23-s + (3.81 − 3.22i)25-s + (0.707 + 0.707i)27-s − 0.894i·29-s + 6.90·31-s + ⋯ |
L(s) = 1 | + (−0.408 + 0.408i)3-s + (0.939 − 0.343i)5-s + (0.00443 − 0.00443i)7-s − 0.333i·9-s − 0.520i·11-s + (−0.657 + 0.657i)13-s + (−0.243 + 0.523i)15-s + (−0.404 + 0.404i)17-s + 1.54·19-s + 0.00361i·21-s + (0.999 + 0.0112i)23-s + (0.763 − 0.645i)25-s + (0.136 + 0.136i)27-s − 0.166i·29-s + 1.23·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.107i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.753599248\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.753599248\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (-2.09 + 0.768i)T \) |
| 23 | \( 1 + (-4.79 - 0.0541i)T \) |
good | 7 | \( 1 + (-0.0117 + 0.0117i)T - 7iT^{2} \) |
| 11 | \( 1 + 1.72iT - 11T^{2} \) |
| 13 | \( 1 + (2.37 - 2.37i)T - 13iT^{2} \) |
| 17 | \( 1 + (1.66 - 1.66i)T - 17iT^{2} \) |
| 19 | \( 1 - 6.73T + 19T^{2} \) |
| 29 | \( 1 + 0.894iT - 29T^{2} \) |
| 31 | \( 1 - 6.90T + 31T^{2} \) |
| 37 | \( 1 + (-0.0575 + 0.0575i)T - 37iT^{2} \) |
| 41 | \( 1 - 0.278T + 41T^{2} \) |
| 43 | \( 1 + (4.31 + 4.31i)T + 43iT^{2} \) |
| 47 | \( 1 + (7.61 + 7.61i)T + 47iT^{2} \) |
| 53 | \( 1 + (-9.64 - 9.64i)T + 53iT^{2} \) |
| 59 | \( 1 + 15.1iT - 59T^{2} \) |
| 61 | \( 1 - 5.91iT - 61T^{2} \) |
| 67 | \( 1 + (-1.99 + 1.99i)T - 67iT^{2} \) |
| 71 | \( 1 - 12.6T + 71T^{2} \) |
| 73 | \( 1 + (-7.55 + 7.55i)T - 73iT^{2} \) |
| 79 | \( 1 + 3.91T + 79T^{2} \) |
| 83 | \( 1 + (-12.2 - 12.2i)T + 83iT^{2} \) |
| 89 | \( 1 - 6.61T + 89T^{2} \) |
| 97 | \( 1 + (9.80 - 9.80i)T - 97iT^{2} \) |
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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
−9.569040731055050843244449401051, −9.050477280398807382520215468945, −8.097992343801898837662233881728, −6.93345230070992315331378158549, −6.26770570558767776744425080476, −5.27166218704857624789685266070, −4.80606680192288946448211818705, −3.53253653068674641684607106560, −2.37180769626786954754545335271, −1.00672766938372658778438746687,
1.04830715562270762871612926880, 2.34312055574950667296944077390, 3.19658390264068415957685094500, 4.89061992073010942104904167026, 5.29482773005270944127581883099, 6.37705143909967770718672965717, 7.03867115220242534897532581420, 7.74662620816615953588345054262, 8.853503077405349369008616659929, 9.839487415448481555643011882927