L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s − 2·7-s + 8-s + 9-s + 10-s − 4·11-s + 12-s − 13-s − 2·14-s + 15-s + 16-s + 18-s − 6·19-s + 20-s − 2·21-s − 4·22-s − 6·23-s + 24-s + 25-s − 26-s + 27-s − 2·28-s + 4·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s + 0.288·12-s − 0.277·13-s − 0.534·14-s + 0.258·15-s + 1/4·16-s + 0.235·18-s − 1.37·19-s + 0.223·20-s − 0.436·21-s − 0.852·22-s − 1.25·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s − 0.377·28-s + 0.742·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.687252160\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.687252160\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−13.51787798666990, −13.18672580150288, −12.75203082401226, −12.45526843203785, −11.88108214964798, −11.12029567238284, −10.66624435482106, −10.24014602452269, −9.664997375620393, −9.478855552301767, −8.492830324826625, −8.134585291009268, −7.820417377661690, −6.900214389726645, −6.568190391065398, −6.170340120884571, −5.356832640193638, −5.094888704131694, −4.250565594764070, −3.892745553049494, −3.159634545304905, −2.527234821373179, −2.303437338572020, −1.528907273038957, −0.3942286368814807,
0.3942286368814807, 1.528907273038957, 2.303437338572020, 2.527234821373179, 3.159634545304905, 3.892745553049494, 4.250565594764070, 5.094888704131694, 5.356832640193638, 6.170340120884571, 6.568190391065398, 6.900214389726645, 7.820417377661690, 8.134585291009268, 8.492830324826625, 9.478855552301767, 9.664997375620393, 10.24014602452269, 10.66624435482106, 11.12029567238284, 11.88108214964798, 12.45526843203785, 12.75203082401226, 13.18672580150288, 13.51787798666990