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 − 6·13-s + 2·14-s + 15-s + 16-s + 4·17-s − 18-s + 19-s − 20-s + 2·21-s − 4·22-s + 4·23-s + 24-s + 25-s + 6·26-s − 27-s − 2·28-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 − 1.66·13-s + 0.534·14-s + 0.258·15-s + 1/4·16-s + 0.970·17-s − 0.235·18-s + 0.229·19-s − 0.223·20-s + 0.436·21-s − 0.852·22-s + 0.834·23-s + 0.204·24-s + 1/5·25-s + 1.17·26-s − 0.192·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7141450331\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7141450331\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 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
−10.71965064877509373359972567112, −9.522328308282647462386870532953, −9.470222020522607664198519044898, −7.956475831183138507617074080251, −7.15887741770707292224397951785, −6.44285077394556366252185445951, −5.29375417302496292014113493094, −4.03167582022396184420156271322, −2.72704646591742030946918279264, −0.858506212559172208583542750856,
0.858506212559172208583542750856, 2.72704646591742030946918279264, 4.03167582022396184420156271322, 5.29375417302496292014113493094, 6.44285077394556366252185445951, 7.15887741770707292224397951785, 7.956475831183138507617074080251, 9.470222020522607664198519044898, 9.522328308282647462386870532953, 10.71965064877509373359972567112