| L(s) = 1 | − 3-s + 7-s + 9-s − 11-s − 2·17-s + 4·19-s − 21-s − 6·23-s − 27-s + 4·29-s + 10·31-s + 33-s + 8·37-s + 6·41-s − 12·43-s − 8·47-s + 49-s + 2·51-s + 6·53-s − 4·57-s − 8·59-s − 2·61-s + 63-s + 8·67-s + 6·69-s − 8·71-s + 8·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.485·17-s + 0.917·19-s − 0.218·21-s − 1.25·23-s − 0.192·27-s + 0.742·29-s + 1.79·31-s + 0.174·33-s + 1.31·37-s + 0.937·41-s − 1.82·43-s − 1.16·47-s + 1/7·49-s + 0.280·51-s + 0.824·53-s − 0.529·57-s − 1.04·59-s − 0.256·61-s + 0.125·63-s + 0.977·67-s + 0.722·69-s − 0.949·71-s + 0.936·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.020502888\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.020502888\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−12.44771177820175, −11.92105742366390, −11.62441084946268, −11.24811805383227, −10.77173899585496, −10.11910996661720, −9.897102743469534, −9.550317395808048, −8.774199173980571, −8.266268960792188, −8.006309270690529, −7.479348640984145, −6.923080208926628, −6.333949526600284, −6.122982245861292, −5.499871830387154, −4.918294215358643, −4.590478151009753, −4.141133233695286, −3.416721490520795, −2.855841012580426, −2.304879475689160, −1.656326796528573, −1.015678672410970, −0.4324198938906397,
0.4324198938906397, 1.015678672410970, 1.656326796528573, 2.304879475689160, 2.855841012580426, 3.416721490520795, 4.141133233695286, 4.590478151009753, 4.918294215358643, 5.499871830387154, 6.122982245861292, 6.333949526600284, 6.923080208926628, 7.479348640984145, 8.006309270690529, 8.266268960792188, 8.774199173980571, 9.550317395808048, 9.897102743469534, 10.11910996661720, 10.77173899585496, 11.24811805383227, 11.62441084946268, 11.92105742366390, 12.44771177820175
