L(s) = 1 | − 3-s + 5-s − 7-s + 9-s − 6·13-s − 15-s − 2·17-s − 4·19-s + 21-s + 4·23-s + 25-s − 27-s − 6·29-s − 35-s − 6·37-s + 6·39-s − 2·41-s + 4·43-s + 45-s + 8·47-s + 49-s + 2·51-s + 2·53-s + 4·57-s + 12·59-s − 6·61-s − 63-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.66·13-s − 0.258·15-s − 0.485·17-s − 0.917·19-s + 0.218·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s − 0.169·35-s − 0.986·37-s + 0.960·39-s − 0.312·41-s + 0.609·43-s + 0.149·45-s + 1.16·47-s + 1/7·49-s + 0.280·51-s + 0.274·53-s + 0.529·57-s + 1.56·59-s − 0.768·61-s − 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.043984976\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.043984976\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−7.78771326537234397428572766309, −7.13594282356795564925351015547, −6.65188756763794875412123642744, −5.82459435706034301214536970935, −5.17132924906787283638744903811, −4.55741521344740139776695940608, −3.65338699205419492308139361350, −2.56061601536455182512634843720, −1.93228573160141279491787242270, −0.51650572393166852625128947290,
0.51650572393166852625128947290, 1.93228573160141279491787242270, 2.56061601536455182512634843720, 3.65338699205419492308139361350, 4.55741521344740139776695940608, 5.17132924906787283638744903811, 5.82459435706034301214536970935, 6.65188756763794875412123642744, 7.13594282356795564925351015547, 7.78771326537234397428572766309