| L(s) = 1 | + 2-s + 1.61·3-s + 4-s + 5-s + 1.61·6-s − 3.61·7-s + 8-s − 0.381·9-s + 10-s − 11-s + 1.61·12-s − 3.61·14-s + 1.61·15-s + 16-s + 0.618·17-s − 0.381·18-s − 7.85·19-s + 20-s − 5.85·21-s − 22-s + 4·23-s + 1.61·24-s + 25-s − 5.47·27-s − 3.61·28-s − 7.70·29-s + 1.61·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.934·3-s + 0.5·4-s + 0.447·5-s + 0.660·6-s − 1.36·7-s + 0.353·8-s − 0.127·9-s + 0.316·10-s − 0.301·11-s + 0.467·12-s − 0.966·14-s + 0.417·15-s + 0.250·16-s + 0.149·17-s − 0.0900·18-s − 1.80·19-s + 0.223·20-s − 1.27·21-s − 0.213·22-s + 0.834·23-s + 0.330·24-s + 0.200·25-s − 1.05·27-s − 0.683·28-s − 1.43·29-s + 0.295·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 3 | \( 1 - 1.61T + 3T^{2} \) |
| 7 | \( 1 + 3.61T + 7T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 0.618T + 17T^{2} \) |
| 19 | \( 1 + 7.85T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 7.70T + 29T^{2} \) |
| 31 | \( 1 + 10.4T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 2.76T + 41T^{2} \) |
| 47 | \( 1 - 0.381T + 47T^{2} \) |
| 53 | \( 1 + 4.14T + 53T^{2} \) |
| 59 | \( 1 + 5.61T + 59T^{2} \) |
| 61 | \( 1 + 5.23T + 61T^{2} \) |
| 67 | \( 1 - 15.4T + 67T^{2} \) |
| 71 | \( 1 - 11.7T + 71T^{2} \) |
| 73 | \( 1 - 9.23T + 73T^{2} \) |
| 79 | \( 1 - 2.85T + 79T^{2} \) |
| 83 | \( 1 - 15.3T + 83T^{2} \) |
| 89 | \( 1 + 15.7T + 89T^{2} \) |
| 97 | \( 1 + 3.52T + 97T^{2} \) |
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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
−7.911960942292277863299208408168, −7.07442084443049231152498644132, −6.44391396701804790951275948958, −5.77114050510170257245805234803, −5.02505668381035075327947835401, −3.78767910476639967300561062028, −3.45671222145974125108833539276, −2.53057822302273096003852121824, −1.91992147001859658351019062368, 0,
1.91992147001859658351019062368, 2.53057822302273096003852121824, 3.45671222145974125108833539276, 3.78767910476639967300561062028, 5.02505668381035075327947835401, 5.77114050510170257245805234803, 6.44391396701804790951275948958, 7.07442084443049231152498644132, 7.911960942292277863299208408168
