| L(s) = 1 | − 3-s − 4.27·5-s + 9-s − 4.27·11-s − 1.27·13-s + 4.27·15-s − 4·17-s + 1.27·19-s + 4·23-s + 13.2·25-s − 27-s − 2.27·29-s − 31-s + 4.27·33-s + 5.27·37-s + 1.27·39-s + 10.5·41-s − 7.27·43-s − 4.27·45-s − 6·47-s + 4·51-s + 1.72·53-s + 18.2·55-s − 1.27·57-s + 6.27·59-s + 10·61-s + 5.45·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.91·5-s + 0.333·9-s − 1.28·11-s − 0.353·13-s + 1.10·15-s − 0.970·17-s + 0.292·19-s + 0.834·23-s + 2.65·25-s − 0.192·27-s − 0.422·29-s − 0.179·31-s + 0.744·33-s + 0.867·37-s + 0.204·39-s + 1.64·41-s − 1.10·43-s − 0.637·45-s − 0.875·47-s + 0.560·51-s + 0.236·53-s + 2.46·55-s − 0.168·57-s + 0.816·59-s + 1.28·61-s + 0.676·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5526642749\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5526642749\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 4.27T + 5T^{2} \) |
| 11 | \( 1 + 4.27T + 11T^{2} \) |
| 13 | \( 1 + 1.27T + 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 - 1.27T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 2.27T + 29T^{2} \) |
| 31 | \( 1 + T + 31T^{2} \) |
| 37 | \( 1 - 5.27T + 37T^{2} \) |
| 41 | \( 1 - 10.5T + 41T^{2} \) |
| 43 | \( 1 + 7.27T + 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 - 1.72T + 53T^{2} \) |
| 59 | \( 1 - 6.27T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 7.27T + 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 - 3.27T + 73T^{2} \) |
| 79 | \( 1 + 3.54T + 79T^{2} \) |
| 83 | \( 1 + 0.274T + 83T^{2} \) |
| 89 | \( 1 - 4.54T + 89T^{2} \) |
| 97 | \( 1 - 16.2T + 97T^{2} \) |
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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
−9.893047682520536524829097232401, −8.773967197862204582857308464921, −7.967648944732108599949944548235, −7.39737911567820883226380929953, −6.64814287949737128465793964331, −5.28667442868940963731636719265, −4.63366197171653645315278411664, −3.72906215364402564707401501441, −2.62305200017329096447616697606, −0.54925869433063225105413337502,
0.54925869433063225105413337502, 2.62305200017329096447616697606, 3.72906215364402564707401501441, 4.63366197171653645315278411664, 5.28667442868940963731636719265, 6.64814287949737128465793964331, 7.39737911567820883226380929953, 7.967648944732108599949944548235, 8.773967197862204582857308464921, 9.893047682520536524829097232401
