L(s) = 1 | − 3-s + 2·5-s − 4·7-s + 9-s − 13-s − 2·15-s + 2·17-s + 8·19-s + 4·21-s − 8·23-s − 25-s − 27-s + 2·29-s − 4·31-s − 8·35-s + 10·37-s + 39-s + 2·41-s − 4·43-s + 2·45-s + 12·47-s + 9·49-s − 2·51-s − 6·53-s − 8·57-s + 2·61-s − 4·63-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.894·5-s − 1.51·7-s + 1/3·9-s − 0.277·13-s − 0.516·15-s + 0.485·17-s + 1.83·19-s + 0.872·21-s − 1.66·23-s − 1/5·25-s − 0.192·27-s + 0.371·29-s − 0.718·31-s − 1.35·35-s + 1.64·37-s + 0.160·39-s + 0.312·41-s − 0.609·43-s + 0.298·45-s + 1.75·47-s + 9/7·49-s − 0.280·51-s − 0.824·53-s − 1.05·57-s + 0.256·61-s − 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.348036396\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.348036396\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 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
−9.405665373957138363971402550297, −8.030822291145243863636280654097, −7.26647645453145107158431203997, −6.41749879814032609293891691510, −5.84499691214238677445756829280, −5.31392621581472689283661321579, −4.03835973199494453539076340423, −3.17373526910944205473955306700, −2.15210120820249014558282717063, −0.74871123248570476997224833420,
0.74871123248570476997224833420, 2.15210120820249014558282717063, 3.17373526910944205473955306700, 4.03835973199494453539076340423, 5.31392621581472689283661321579, 5.84499691214238677445756829280, 6.41749879814032609293891691510, 7.26647645453145107158431203997, 8.030822291145243863636280654097, 9.405665373957138363971402550297