| L(s) = 1 | − 3-s − 3·7-s + 9-s − 2·11-s − 6·13-s − 2·17-s − 19-s + 3·21-s + 3·23-s − 27-s + 6·29-s − 2·31-s + 2·33-s − 2·37-s + 6·39-s − 5·41-s + 8·43-s + 8·47-s + 2·49-s + 2·51-s − 5·53-s + 57-s − 12·59-s + 8·61-s − 3·63-s + 3·67-s − 3·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.13·7-s + 1/3·9-s − 0.603·11-s − 1.66·13-s − 0.485·17-s − 0.229·19-s + 0.654·21-s + 0.625·23-s − 0.192·27-s + 1.11·29-s − 0.359·31-s + 0.348·33-s − 0.328·37-s + 0.960·39-s − 0.780·41-s + 1.21·43-s + 1.16·47-s + 2/7·49-s + 0.280·51-s − 0.686·53-s + 0.132·57-s − 1.56·59-s + 1.02·61-s − 0.377·63-s + 0.366·67-s − 0.361·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5723645680\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5723645680\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 167 | \( 1 - T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−12.90501587056094, −12.55447746021287, −12.24857474399560, −11.79131685704314, −11.09395001459651, −10.67174665821182, −10.26573870065805, −9.838139033831278, −9.273282112192651, −9.073557207216543, −8.175307198553501, −7.801214309424986, −7.100916711390566, −6.830954130365116, −6.424543356892963, −5.694647136688316, −5.325758133658991, −4.709105990789471, −4.372900691115315, −3.572810141584136, −2.967913380361313, −2.510128402307355, −1.960841671885984, −0.9257858299126178, −0.2622575766230507,
0.2622575766230507, 0.9257858299126178, 1.960841671885984, 2.510128402307355, 2.967913380361313, 3.572810141584136, 4.372900691115315, 4.709105990789471, 5.325758133658991, 5.694647136688316, 6.424543356892963, 6.830954130365116, 7.100916711390566, 7.801214309424986, 8.175307198553501, 9.073557207216543, 9.273282112192651, 9.838139033831278, 10.26573870065805, 10.67174665821182, 11.09395001459651, 11.79131685704314, 12.24857474399560, 12.55447746021287, 12.90501587056094
