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
−8.093918147859020171507176024935, −7.54589818118301820357274453582, −6.67519169143584120937097545513, −6.31409620129345260214814525694, −5.07841413722666243729852725725, −4.40825594732626851832215490337, −3.37721044627627627731585389728, −2.59924538469102992238637666025, −1.17429122779692212625963207886, −0.35720249118236664948040314366,
1.82518497361076507274090842118, 2.50410773825798860514505138230, 3.57119675450040322613192029112, 4.52526555564160521048747049391, 5.23750171066785105713896496118, 5.87322303603784433446890478681, 6.75928121739465375726192172993, 7.61971616911127348160666198782, 8.504995662132083264407473647956, 9.092073775247950372142565643745