| L(s) = 1 | + 3·3-s − 2·7-s + 6·9-s + 11-s − 4·13-s − 5·17-s + 19-s − 6·21-s + 2·23-s + 9·27-s − 8·29-s + 10·31-s + 3·33-s + 6·37-s − 12·39-s − 3·41-s − 4·43-s − 4·47-s − 3·49-s − 15·51-s − 6·53-s + 3·57-s + 8·59-s + 10·61-s − 12·63-s + 67-s + 6·69-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.755·7-s + 2·9-s + 0.301·11-s − 1.10·13-s − 1.21·17-s + 0.229·19-s − 1.30·21-s + 0.417·23-s + 1.73·27-s − 1.48·29-s + 1.79·31-s + 0.522·33-s + 0.986·37-s − 1.92·39-s − 0.468·41-s − 0.609·43-s − 0.583·47-s − 3/7·49-s − 2.10·51-s − 0.824·53-s + 0.397·57-s + 1.04·59-s + 1.28·61-s − 1.51·63-s + 0.122·67-s + 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.808107531\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.808107531\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 13 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−12.91670582621015996055700148464, −11.57805678356630223088027639579, −10.02244514988645342042396669245, −9.447015792876853394703046750020, −8.567472421484776156138109958322, −7.52156329683392804904040430602, −6.57858828761327407864935206739, −4.60235027296278679518317581481, −3.34359627566450606358537194196, −2.26917003136356429029887098788,
2.26917003136356429029887098788, 3.34359627566450606358537194196, 4.60235027296278679518317581481, 6.57858828761327407864935206739, 7.52156329683392804904040430602, 8.567472421484776156138109958322, 9.447015792876853394703046750020, 10.02244514988645342042396669245, 11.57805678356630223088027639579, 12.91670582621015996055700148464
