| L(s) = 1 | + 3·5-s − 3·11-s − 2·13-s + 3·17-s − 19-s + 3·23-s + 4·25-s + 6·29-s − 7·31-s − 37-s + 6·41-s + 4·43-s + 9·47-s − 3·53-s − 9·55-s − 9·59-s + 61-s − 6·65-s + 7·67-s + 73-s + 13·79-s − 12·83-s + 9·85-s + 15·89-s − 3·95-s + 10·97-s + 15·101-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 0.904·11-s − 0.554·13-s + 0.727·17-s − 0.229·19-s + 0.625·23-s + 4/5·25-s + 1.11·29-s − 1.25·31-s − 0.164·37-s + 0.937·41-s + 0.609·43-s + 1.31·47-s − 0.412·53-s − 1.21·55-s − 1.17·59-s + 0.128·61-s − 0.744·65-s + 0.855·67-s + 0.117·73-s + 1.46·79-s − 1.31·83-s + 0.976·85-s + 1.58·89-s − 0.307·95-s + 1.01·97-s + 1.49·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.517777486\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.517777486\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−7.79471766294442238351763450285, −7.33030683285885341431588133268, −6.40825653451897138853524205007, −5.78984331009086520334042076467, −5.22514419219416143835802109221, −4.57537906789274455198829131722, −3.39490893308522303308712327096, −2.56837123410425775655446467441, −1.96218931226981980488959497856, −0.804522886912069846049191220250,
0.804522886912069846049191220250, 1.96218931226981980488959497856, 2.56837123410425775655446467441, 3.39490893308522303308712327096, 4.57537906789274455198829131722, 5.22514419219416143835802109221, 5.78984331009086520334042076467, 6.40825653451897138853524205007, 7.33030683285885341431588133268, 7.79471766294442238351763450285
