L(s) = 1 | + 1.93i·2-s − 2.73·4-s − 3.34i·8-s − 0.517i·11-s + 3.73·16-s + 0.999·22-s + 1.41i·23-s + 25-s + 1.41i·29-s + 3.86i·32-s + 1.73·37-s − 1.73·43-s + 1.41i·44-s − 2.73·46-s + 1.93i·50-s + ⋯ |
L(s) = 1 | + 1.93i·2-s − 2.73·4-s − 3.34i·8-s − 0.517i·11-s + 3.73·16-s + 0.999·22-s + 1.41i·23-s + 25-s + 1.41i·29-s + 3.86i·32-s + 1.73·37-s − 1.73·43-s + 1.41i·44-s − 2.73·46-s + 1.93i·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9553796115\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9553796115\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 1.93iT - T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 11 | \( 1 + 0.517iT - T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - 1.41iT - T^{2} \) |
| 29 | \( 1 - 1.41iT - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - 1.73T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 1.73T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - 0.517iT - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 - T + T^{2} \) |
| 71 | \( 1 - 1.93iT - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−8.683054102841146517862818277031, −8.190905973288598584106637796437, −7.39149750615712762349894314834, −6.86786957467052075836182543472, −6.13133776019401321388510169051, −5.42421416350708734240397328349, −4.87394311557662320559463637253, −3.93528692086243315427971413839, −3.11028620019379152938681199562, −1.15120527358256937590115026030,
0.65498837616697276601980948779, 1.87254659880856011903497559680, 2.60679355840232943341171392509, 3.38336366409589532500310773252, 4.43821875281113136516768523565, 4.68668345871981838052957328624, 5.78296006982442827819035302821, 6.77882796565483789517463269972, 7.997797489445745322720673568909, 8.463311993714024538457640353503