| L(s) = 1 | + 1.12·2-s + 2.45·3-s − 0.728·4-s − 2.17·5-s + 2.77·6-s − 3.35·7-s − 3.07·8-s + 3.03·9-s − 2.45·10-s − 11-s − 1.79·12-s − 3.30·13-s − 3.77·14-s − 5.34·15-s − 2.01·16-s − 17-s + 3.42·18-s + 3.24·19-s + 1.58·20-s − 8.23·21-s − 1.12·22-s − 1.45·23-s − 7.56·24-s − 0.275·25-s − 3.72·26-s + 0.0942·27-s + 2.44·28-s + ⋯ |
| L(s) = 1 | + 0.797·2-s + 1.41·3-s − 0.364·4-s − 0.972·5-s + 1.13·6-s − 1.26·7-s − 1.08·8-s + 1.01·9-s − 0.775·10-s − 0.301·11-s − 0.516·12-s − 0.916·13-s − 1.00·14-s − 1.37·15-s − 0.503·16-s − 0.242·17-s + 0.807·18-s + 0.745·19-s + 0.354·20-s − 1.79·21-s − 0.240·22-s − 0.303·23-s − 1.54·24-s − 0.0551·25-s − 0.731·26-s + 0.0181·27-s + 0.461·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.939404962\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.939404962\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 - 1.12T + 2T^{2} \) |
| 3 | \( 1 - 2.45T + 3T^{2} \) |
| 5 | \( 1 + 2.17T + 5T^{2} \) |
| 7 | \( 1 + 3.35T + 7T^{2} \) |
| 13 | \( 1 + 3.30T + 13T^{2} \) |
| 19 | \( 1 - 3.24T + 19T^{2} \) |
| 23 | \( 1 + 1.45T + 23T^{2} \) |
| 29 | \( 1 - 5.35T + 29T^{2} \) |
| 31 | \( 1 - 3.42T + 31T^{2} \) |
| 37 | \( 1 - 2.58T + 37T^{2} \) |
| 41 | \( 1 - 9.27T + 41T^{2} \) |
| 47 | \( 1 + 5.00T + 47T^{2} \) |
| 53 | \( 1 - 2.16T + 53T^{2} \) |
| 59 | \( 1 + 6.10T + 59T^{2} \) |
| 61 | \( 1 - 0.705T + 61T^{2} \) |
| 67 | \( 1 - 1.74T + 67T^{2} \) |
| 71 | \( 1 - 2.99T + 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 - 9.44T + 79T^{2} \) |
| 83 | \( 1 + 9.81T + 83T^{2} \) |
| 89 | \( 1 - 6.20T + 89T^{2} \) |
| 97 | \( 1 + 9.69T + 97T^{2} \) |
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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
−7.80908657021911669134022863606, −7.32571989093043437529506450588, −6.43946025532362360504762758550, −5.70554428146661334933082693855, −4.64613144133494441109608867662, −4.21067994127326262623310641183, −3.34450501312466922539374705380, −3.05578829101714686340822206630, −2.33822658718482274751007070921, −0.53507956205337962249803961856,
0.53507956205337962249803961856, 2.33822658718482274751007070921, 3.05578829101714686340822206630, 3.34450501312466922539374705380, 4.21067994127326262623310641183, 4.64613144133494441109608867662, 5.70554428146661334933082693855, 6.43946025532362360504762758550, 7.32571989093043437529506450588, 7.80908657021911669134022863606
