L(s) = 1 | + i·3-s + 4i·7-s − 9-s + 4i·11-s − 2·13-s + (1 + 4i)17-s − 4·19-s − 4·21-s + 4i·23-s + 5·25-s − i·27-s + 4i·31-s − 4·33-s − 8i·37-s − 2i·39-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 1.51i·7-s − 0.333·9-s + 1.20i·11-s − 0.554·13-s + (0.242 + 0.970i)17-s − 0.917·19-s − 0.872·21-s + 0.834i·23-s + 25-s − 0.192i·27-s + 0.718i·31-s − 0.696·33-s − 1.31i·37-s − 0.320i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 + 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.114495186\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.114495186\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 17 | \( 1 + (-1 - 4i)T \) |
good | 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 - 4iT - 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 4iT - 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 + 8iT - 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 8iT - 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 + 12iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 4iT - 79T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 16iT - 97T^{2} \) |
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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
−9.092073775247950372142565643745, −8.504995662132083264407473647956, −7.61971616911127348160666198782, −6.75928121739465375726192172993, −5.87322303603784433446890478681, −5.23750171066785105713896496118, −4.52526555564160521048747049391, −3.57119675450040322613192029112, −2.50410773825798860514505138230, −1.82518497361076507274090842118,
0.35720249118236664948040314366, 1.17429122779692212625963207886, 2.59924538469102992238637666025, 3.37721044627627627731585389728, 4.40825594732626851832215490337, 5.07841413722666243729852725725, 6.31409620129345260214814525694, 6.67519169143584120937097545513, 7.54589818118301820357274453582, 8.093918147859020171507176024935