| L(s) = 1 | + 0.483·2-s + 3-s − 1.76·4-s + 3.13·5-s + 0.483·6-s − 7-s − 1.82·8-s + 9-s + 1.51·10-s + 1.81·11-s − 1.76·12-s − 5.27·13-s − 0.483·14-s + 3.13·15-s + 2.65·16-s − 5.52·17-s + 0.483·18-s + 5.67·19-s − 5.54·20-s − 21-s + 0.876·22-s + 8.07·23-s − 1.82·24-s + 4.84·25-s − 2.54·26-s + 27-s + 1.76·28-s + ⋯ |
| L(s) = 1 | + 0.341·2-s + 0.577·3-s − 0.883·4-s + 1.40·5-s + 0.197·6-s − 0.377·7-s − 0.643·8-s + 0.333·9-s + 0.479·10-s + 0.546·11-s − 0.509·12-s − 1.46·13-s − 0.129·14-s + 0.810·15-s + 0.663·16-s − 1.34·17-s + 0.113·18-s + 1.30·19-s − 1.23·20-s − 0.218·21-s + 0.186·22-s + 1.68·23-s − 0.371·24-s + 0.969·25-s − 0.499·26-s + 0.192·27-s + 0.333·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.931474759\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.931474759\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 - 0.483T + 2T^{2} \) |
| 5 | \( 1 - 3.13T + 5T^{2} \) |
| 11 | \( 1 - 1.81T + 11T^{2} \) |
| 13 | \( 1 + 5.27T + 13T^{2} \) |
| 17 | \( 1 + 5.52T + 17T^{2} \) |
| 19 | \( 1 - 5.67T + 19T^{2} \) |
| 23 | \( 1 - 8.07T + 23T^{2} \) |
| 29 | \( 1 - 0.836T + 29T^{2} \) |
| 31 | \( 1 + 1.70T + 31T^{2} \) |
| 37 | \( 1 - 1.51T + 37T^{2} \) |
| 41 | \( 1 + 2.67T + 41T^{2} \) |
| 43 | \( 1 + 1.94T + 43T^{2} \) |
| 47 | \( 1 - 4.46T + 47T^{2} \) |
| 53 | \( 1 - 0.415T + 53T^{2} \) |
| 59 | \( 1 - 4.01T + 59T^{2} \) |
| 61 | \( 1 + 12.5T + 61T^{2} \) |
| 67 | \( 1 - 6.17T + 67T^{2} \) |
| 71 | \( 1 - 11.7T + 71T^{2} \) |
| 73 | \( 1 - 4.70T + 73T^{2} \) |
| 79 | \( 1 + 1.31T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 + 1.37T + 89T^{2} \) |
| 97 | \( 1 - 3.85T + 97T^{2} \) |
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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
−7.83154249877104500538757147609, −6.98010656422999016596409172195, −6.51006702595997172601575655048, −5.53355816246576341855874688829, −5.04455161342230353184532064230, −4.43233924765926699948562523061, −3.39387991331800136960400704381, −2.74307345026594162273956920615, −1.95330827220270665253294047888, −0.78789347761844056916239428198,
0.78789347761844056916239428198, 1.95330827220270665253294047888, 2.74307345026594162273956920615, 3.39387991331800136960400704381, 4.43233924765926699948562523061, 5.04455161342230353184532064230, 5.53355816246576341855874688829, 6.51006702595997172601575655048, 6.98010656422999016596409172195, 7.83154249877104500538757147609
