L(s) = 1 | + 2-s − 3-s + 4-s + 2·5-s − 6-s + 8-s + 9-s + 2·10-s − 4·11-s − 12-s + 2·13-s − 2·15-s + 16-s − 6·17-s + 18-s − 19-s + 2·20-s − 4·22-s − 4·23-s − 24-s − 25-s + 2·26-s − 27-s − 2·29-s − 2·30-s + 4·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.632·10-s − 1.20·11-s − 0.288·12-s + 0.554·13-s − 0.516·15-s + 1/4·16-s − 1.45·17-s + 0.235·18-s − 0.229·19-s + 0.447·20-s − 0.852·22-s − 0.834·23-s − 0.204·24-s − 1/5·25-s + 0.392·26-s − 0.192·27-s − 0.371·29-s − 0.365·30-s + 0.718·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.387723731\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.387723731\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 + T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.26471264838449267214147905866, −12.97088967028240784250602792338, −11.47795618604630761722336104692, −10.67649349923181634582213825679, −9.600878448141019797397427731260, −7.987570621007900466899230715503, −6.46925034993373182658971924423, −5.66755609194805589043695166783, −4.39790054298019034493919320980, −2.33973473634575001863443152397,
2.33973473634575001863443152397, 4.39790054298019034493919320980, 5.66755609194805589043695166783, 6.46925034993373182658971924423, 7.987570621007900466899230715503, 9.600878448141019797397427731260, 10.67649349923181634582213825679, 11.47795618604630761722336104692, 12.97088967028240784250602792338, 13.26471264838449267214147905866