| L(s) = 1 | + 3·5-s − 7-s + 3·13-s + 6·17-s + 19-s − 4·23-s + 4·25-s + 5·29-s + 4·31-s − 3·35-s + 5·37-s + 2·41-s + 6·43-s − 47-s + 49-s + 2·53-s + 7·59-s + 14·61-s + 9·65-s − 67-s + 6·71-s + 17·73-s + 2·79-s − 14·83-s + 18·85-s + 6·89-s − 3·91-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 0.377·7-s + 0.832·13-s + 1.45·17-s + 0.229·19-s − 0.834·23-s + 4/5·25-s + 0.928·29-s + 0.718·31-s − 0.507·35-s + 0.821·37-s + 0.312·41-s + 0.914·43-s − 0.145·47-s + 1/7·49-s + 0.274·53-s + 0.911·59-s + 1.79·61-s + 1.11·65-s − 0.122·67-s + 0.712·71-s + 1.98·73-s + 0.225·79-s − 1.53·83-s + 1.95·85-s + 0.635·89-s − 0.314·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.947396447\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.947396447\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 17 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−13.67973704392012, −13.08056875234464, −12.63062799866005, −12.26097568434372, −11.53908269034286, −11.20923385001168, −10.37871919812353, −10.05307099467531, −9.838538369449798, −9.247638704131716, −8.703873830835515, −8.124281204393306, −7.724616897173829, −6.957800228544426, −6.381646293664119, −6.074541800910392, −5.539799021996774, −5.171502702796442, −4.329365572598832, −3.737658367118295, −3.165337674276545, −2.470564807901006, −2.032090391256499, −1.086185568412033, −0.8016709650142065,
0.8016709650142065, 1.086185568412033, 2.032090391256499, 2.470564807901006, 3.165337674276545, 3.737658367118295, 4.329365572598832, 5.171502702796442, 5.539799021996774, 6.074541800910392, 6.381646293664119, 6.957800228544426, 7.724616897173829, 8.124281204393306, 8.703873830835515, 9.247638704131716, 9.838538369449798, 10.05307099467531, 10.37871919812353, 11.20923385001168, 11.53908269034286, 12.26097568434372, 12.63062799866005, 13.08056875234464, 13.67973704392012
