L(s) = 1 | − i·3-s − 1.41i·5-s − 1.41·7-s − 9-s − 1.41·15-s + 1.41i·21-s − 1.00·25-s + i·27-s − 1.41i·29-s − 1.41·31-s + 2.00i·35-s + 1.41i·45-s + 1.00·49-s + 1.41i·53-s − 2i·59-s + ⋯ |
L(s) = 1 | − i·3-s − 1.41i·5-s − 1.41·7-s − 9-s − 1.41·15-s + 1.41i·21-s − 1.00·25-s + i·27-s − 1.41i·29-s − 1.41·31-s + 2.00i·35-s + 1.41i·45-s + 1.00·49-s + 1.41i·53-s − 2i·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6306205026\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6306205026\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
good | 5 | \( 1 + 1.41iT - T^{2} \) |
| 7 | \( 1 + 1.41T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + 1.41iT - T^{2} \) |
| 31 | \( 1 + 1.41T + T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - 1.41iT - T^{2} \) |
| 59 | \( 1 + 2iT - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 - 1.41T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + 2T + T^{2} \) |
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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
−9.248646551286118497724882773482, −8.444171678897054832532475526433, −7.70108622366752682736378142173, −6.79086485975441440948786605513, −6.01164636642245522598555766092, −5.32786036996171564093830852157, −4.14019518564257500800259096714, −3.05760073405233980040513067642, −1.83323515450409766242867810768, −0.47723753458551672793786135729,
2.48685273683211512943679477018, 3.33399801338009607358460732162, 3.77209691869147099429817644126, 5.15416656696291014008391082304, 6.06721125677170315531463711802, 6.73824302184038454308079115223, 7.46336340747380044247858058757, 8.737814088621360681779959284018, 9.413489807095613835543953171602, 10.11694378337472505584587902867