L(s) = 1 | + i·2-s − 4-s − 5-s + (0.707 + 0.707i)7-s − i·8-s − i·10-s + 1.41i·11-s − 1.41i·13-s + (−0.707 + 0.707i)14-s + 16-s + 2i·19-s + 20-s − 1.41·22-s + 25-s + 1.41·26-s + ⋯ |
L(s) = 1 | + i·2-s − 4-s − 5-s + (0.707 + 0.707i)7-s − i·8-s − i·10-s + 1.41i·11-s − 1.41i·13-s + (−0.707 + 0.707i)14-s + 16-s + 2i·19-s + 20-s − 1.41·22-s + 25-s + 1.41·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7342437925\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7342437925\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
good | 11 | \( 1 - 1.41iT - T^{2} \) |
| 13 | \( 1 + 1.41iT - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - 2iT - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + 1.41T + T^{2} \) |
| 41 | \( 1 + 1.41T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 - 2iT - T^{2} \) |
| 59 | \( 1 + 1.41T + T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 1.41T + T^{2} \) |
| 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
−9.166072280088095271434326847981, −8.410713731659049451616985276962, −7.78768671335693523974565021520, −7.48773589176896934941046920696, −6.41354280247206214196555895685, −5.49082437436569933668396032790, −4.91782764199582684529951034692, −4.06794579293373820021032246407, −3.16971854292402273983814236694, −1.57802405836628931216191429552,
0.51681731969362598700025945050, 1.78869660497588540143984134110, 3.09953377239775002337465639595, 3.76870990047337264050644278202, 4.62178890907162209606281623611, 5.13490221975061744412791987016, 6.59587087375319018966924779524, 7.27807329331072179804674566090, 8.337335784416169405027001675980, 8.670642602498789473434608025324