| L(s) = 1 | + 3-s + 9-s + 4·11-s + 6·13-s − 2·17-s − 4·19-s + 8·23-s + 27-s − 6·29-s − 31-s + 4·33-s − 2·37-s + 6·39-s + 10·41-s − 4·43-s − 7·49-s − 2·51-s − 10·53-s − 4·57-s + 12·59-s + 2·61-s − 4·67-s + 8·69-s − 2·73-s + 81-s + 4·83-s − 6·87-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 1.20·11-s + 1.66·13-s − 0.485·17-s − 0.917·19-s + 1.66·23-s + 0.192·27-s − 1.11·29-s − 0.179·31-s + 0.696·33-s − 0.328·37-s + 0.960·39-s + 1.56·41-s − 0.609·43-s − 49-s − 0.280·51-s − 1.37·53-s − 0.529·57-s + 1.56·59-s + 0.256·61-s − 0.488·67-s + 0.963·69-s − 0.234·73-s + 1/9·81-s + 0.439·83-s − 0.643·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 148800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 148800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.103091290\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.103091290\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−13.36083657083494, −12.87326116911853, −12.66374450163299, −11.87028902025433, −11.23687315036848, −11.00710094477164, −10.73479505131721, −9.750605205958484, −9.442390218192240, −8.972237685305572, −8.502244922868249, −8.265148109630399, −7.460402761666922, −6.802363746033081, −6.672683352782682, −5.953679709204852, −5.518278675935137, −4.633017474771173, −4.219682652129055, −3.681483851641797, −3.256850674636205, −2.571988164000826, −1.706821221779275, −1.407112371904208, −0.5875554740352553,
0.5875554740352553, 1.407112371904208, 1.706821221779275, 2.571988164000826, 3.256850674636205, 3.681483851641797, 4.219682652129055, 4.633017474771173, 5.518278675935137, 5.953679709204852, 6.672683352782682, 6.802363746033081, 7.460402761666922, 8.265148109630399, 8.502244922868249, 8.972237685305572, 9.442390218192240, 9.750605205958484, 10.73479505131721, 11.00710094477164, 11.23687315036848, 11.87028902025433, 12.66374450163299, 12.87326116911853, 13.36083657083494
