L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s + 4·7-s − 8-s + 9-s − 10-s − 12-s + 2·13-s − 4·14-s − 15-s + 16-s − 18-s − 4·19-s + 20-s − 4·21-s + 24-s + 25-s − 2·26-s − 27-s + 4·28-s + 6·29-s + 30-s − 8·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s + 0.554·13-s − 1.06·14-s − 0.258·15-s + 1/4·16-s − 0.235·18-s − 0.917·19-s + 0.223·20-s − 0.872·21-s + 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s + 0.755·28-s + 1.11·29-s + 0.182·30-s − 1.43·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.625933732\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.625933732\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 17 | \( 1 \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 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
−7.79711986387068141575976779625, −7.19170400608151927655187626887, −6.37150043364000269525301859690, −5.82487279118675041342104526304, −5.01944387438421311944798675589, −4.46472359494342999265261204391, −3.45323495695880355625020856878, −2.19891095670466256908754466292, −1.67422833535095353270525529506, −0.74314124481069084021545695805,
0.74314124481069084021545695805, 1.67422833535095353270525529506, 2.19891095670466256908754466292, 3.45323495695880355625020856878, 4.46472359494342999265261204391, 5.01944387438421311944798675589, 5.82487279118675041342104526304, 6.37150043364000269525301859690, 7.19170400608151927655187626887, 7.79711986387068141575976779625