| L(s) = 1 | + 3·5-s + 6·11-s − 2·13-s + 7·17-s + 2·19-s + 4·23-s + 4·25-s + 2·29-s + 2·31-s − 3·37-s − 3·41-s − 11·43-s + 7·47-s + 12·53-s + 18·55-s + 3·59-s + 10·61-s − 6·65-s − 4·67-s + 12·71-s − 16·73-s − 7·79-s − 7·83-s + 21·85-s + 10·89-s + 6·95-s − 2·97-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 1.80·11-s − 0.554·13-s + 1.69·17-s + 0.458·19-s + 0.834·23-s + 4/5·25-s + 0.371·29-s + 0.359·31-s − 0.493·37-s − 0.468·41-s − 1.67·43-s + 1.02·47-s + 1.64·53-s + 2.42·55-s + 0.390·59-s + 1.28·61-s − 0.744·65-s − 0.488·67-s + 1.42·71-s − 1.87·73-s − 0.787·79-s − 0.768·83-s + 2.27·85-s + 1.05·89-s + 0.615·95-s − 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.736313812\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.736313812\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−16.76566463632620, −16.23036241776455, −15.21734594905397, −14.65590497453850, −14.31437440206503, −13.76435707777335, −13.26168395320319, −12.46128533284201, −11.87112521382819, −11.59510011495935, −10.48415351135515, −9.971684406653241, −9.660195668831153, −8.948213506887601, −8.456719364428045, −7.408201746378593, −6.856384719610558, −6.295355476019585, −5.495114412556691, −5.156884315979007, −4.084485814407015, −3.357539135592577, −2.538555673041712, −1.533009212796695, −1.033222168652044,
1.033222168652044, 1.533009212796695, 2.538555673041712, 3.357539135592577, 4.084485814407015, 5.156884315979007, 5.495114412556691, 6.295355476019585, 6.856384719610558, 7.408201746378593, 8.456719364428045, 8.948213506887601, 9.660195668831153, 9.971684406653241, 10.48415351135515, 11.59510011495935, 11.87112521382819, 12.46128533284201, 13.26168395320319, 13.76435707777335, 14.31437440206503, 14.65590497453850, 15.21734594905397, 16.23036241776455, 16.76566463632620
