| L(s) = 1 | + 3-s − 2·5-s + 7-s + 9-s − 2·11-s − 6·13-s − 2·15-s − 4·17-s − 19-s + 21-s + 4·23-s − 25-s + 27-s − 2·29-s + 6·31-s − 2·33-s − 2·35-s − 4·37-s − 6·39-s + 6·41-s + 4·43-s − 2·45-s + 6·47-s + 49-s − 4·51-s + 6·53-s + 4·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s − 0.603·11-s − 1.66·13-s − 0.516·15-s − 0.970·17-s − 0.229·19-s + 0.218·21-s + 0.834·23-s − 1/5·25-s + 0.192·27-s − 0.371·29-s + 1.07·31-s − 0.348·33-s − 0.338·35-s − 0.657·37-s − 0.960·39-s + 0.937·41-s + 0.609·43-s − 0.298·45-s + 0.875·47-s + 1/7·49-s − 0.560·51-s + 0.824·53-s + 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.472569087\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.472569087\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−7.933382096844684412181512116895, −7.40888146588395100813109468459, −6.96997012027698172529518161901, −5.87569393768260284392444874683, −4.84113037891117783609967249099, −4.52853348527522193774317397968, −3.61885352408121174777420103667, −2.64977241993461444574708554888, −2.12394694083224686946305813040, −0.58329832351649993591138425084,
0.58329832351649993591138425084, 2.12394694083224686946305813040, 2.64977241993461444574708554888, 3.61885352408121174777420103667, 4.52853348527522193774317397968, 4.84113037891117783609967249099, 5.87569393768260284392444874683, 6.96997012027698172529518161901, 7.40888146588395100813109468459, 7.933382096844684412181512116895
