L(s) = 1 | + 2-s − 3.26·3-s + 4-s + 1.58·5-s − 3.26·6-s + 8-s + 7.63·9-s + 1.58·10-s − 0.654·11-s − 3.26·12-s − 3.08·13-s − 5.16·15-s + 16-s − 2.81·17-s + 7.63·18-s + 0.824·19-s + 1.58·20-s − 0.654·22-s + 7.66·23-s − 3.26·24-s − 2.48·25-s − 3.08·26-s − 15.1·27-s − 7.91·29-s − 5.16·30-s − 1.00·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.88·3-s + 0.5·4-s + 0.708·5-s − 1.33·6-s + 0.353·8-s + 2.54·9-s + 0.501·10-s − 0.197·11-s − 0.941·12-s − 0.854·13-s − 1.33·15-s + 0.250·16-s − 0.682·17-s + 1.79·18-s + 0.189·19-s + 0.354·20-s − 0.139·22-s + 1.59·23-s − 0.665·24-s − 0.497·25-s − 0.604·26-s − 2.90·27-s − 1.46·29-s − 0.943·30-s − 0.180·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
good | 3 | \( 1 + 3.26T + 3T^{2} \) |
| 5 | \( 1 - 1.58T + 5T^{2} \) |
| 11 | \( 1 + 0.654T + 11T^{2} \) |
| 13 | \( 1 + 3.08T + 13T^{2} \) |
| 17 | \( 1 + 2.81T + 17T^{2} \) |
| 19 | \( 1 - 0.824T + 19T^{2} \) |
| 23 | \( 1 - 7.66T + 23T^{2} \) |
| 29 | \( 1 + 7.91T + 29T^{2} \) |
| 31 | \( 1 + 1.00T + 31T^{2} \) |
| 37 | \( 1 + 8.23T + 37T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 + 6.45T + 47T^{2} \) |
| 53 | \( 1 + 9.14T + 53T^{2} \) |
| 59 | \( 1 - 4.13T + 59T^{2} \) |
| 61 | \( 1 - 2.96T + 61T^{2} \) |
| 67 | \( 1 + 5.53T + 67T^{2} \) |
| 71 | \( 1 - 8.52T + 71T^{2} \) |
| 73 | \( 1 - 1.19T + 73T^{2} \) |
| 79 | \( 1 - 0.723T + 79T^{2} \) |
| 83 | \( 1 + 16.4T + 83T^{2} \) |
| 89 | \( 1 - 0.882T + 89T^{2} \) |
| 97 | \( 1 - 3.48T + 97T^{2} \) |
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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
−7.56424659163507203332667163246, −7.08330197081779009591291460523, −6.39269155198135527131064508678, −5.71412308485935851359083124717, −5.16224702287845385240440546307, −4.68547797802136435647245682566, −3.68658505640268780638066848346, −2.34154693298182209350350373075, −1.38920706317094944295698130818, 0,
1.38920706317094944295698130818, 2.34154693298182209350350373075, 3.68658505640268780638066848346, 4.68547797802136435647245682566, 5.16224702287845385240440546307, 5.71412308485935851359083124717, 6.39269155198135527131064508678, 7.08330197081779009591291460523, 7.56424659163507203332667163246