L(s) = 1 | + (0.707 − 0.707i)5-s − 2i·13-s − 1.41·17-s − 1.00i·25-s − 1.41i·29-s + 1.41i·41-s + 49-s − 1.41·53-s + 2·61-s + (−1.41 − 1.41i)65-s + 2i·73-s + (−1.00 + 1.00i)85-s − 1.41i·89-s − 2i·97-s + 1.41i·101-s + ⋯ |
L(s) = 1 | + (0.707 − 0.707i)5-s − 2i·13-s − 1.41·17-s − 1.00i·25-s − 1.41i·29-s + 1.41i·41-s + 49-s − 1.41·53-s + 2·61-s + (−1.41 − 1.41i)65-s + 2i·73-s + (−1.00 + 1.00i)85-s − 1.41i·89-s − 2i·97-s + 1.41i·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.250272320\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.250272320\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.707 + 0.707i)T \) |
good | 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + 2iT - T^{2} \) |
| 17 | \( 1 + 1.41T + T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + 1.41iT - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - 1.41iT - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + 1.41T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 2T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 2iT - T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + 1.41iT - T^{2} \) |
| 97 | \( 1 + 2iT - T^{2} \) |
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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
−8.597406698164618218530687850489, −8.272457406295000043022029540664, −7.35194979139766840929829953786, −6.30687507491367460525312548754, −5.74619372932479945664830337675, −4.96246540582496780540651216639, −4.19069659732807037582480196262, −2.96741114604231083956354136386, −2.13073161634101885743960030280, −0.77243171940817987685437170764,
1.74922808533727725495448654560, 2.35494142730473124917319451416, 3.55450206630268070716617230156, 4.41745089041717738587916943047, 5.27831601646393699416224068562, 6.31881907363701022271659036266, 6.79357097658849185537644977339, 7.32779912983168513973858306075, 8.646350057116396230430802559296, 9.113097000047503054415477516749