| L(s) = 1 | − 2.42·2-s + 3.86·4-s + 1.62·5-s + 2.13·7-s − 4.52·8-s − 3.93·10-s − 11-s − 5.27·13-s − 5.18·14-s + 3.22·16-s + 5.69·17-s − 3.03·19-s + 6.27·20-s + 2.42·22-s + 4.60·23-s − 2.36·25-s + 12.7·26-s + 8.27·28-s + 8.32·29-s + 2.27·31-s + 1.24·32-s − 13.7·34-s + 3.47·35-s + 0.699·37-s + 7.35·38-s − 7.34·40-s − 9.03·41-s + ⋯ |
| L(s) = 1 | − 1.71·2-s + 1.93·4-s + 0.725·5-s + 0.808·7-s − 1.59·8-s − 1.24·10-s − 0.301·11-s − 1.46·13-s − 1.38·14-s + 0.805·16-s + 1.38·17-s − 0.696·19-s + 1.40·20-s + 0.516·22-s + 0.959·23-s − 0.473·25-s + 2.50·26-s + 1.56·28-s + 1.54·29-s + 0.407·31-s + 0.220·32-s − 2.36·34-s + 0.586·35-s + 0.115·37-s + 1.19·38-s − 1.16·40-s − 1.41·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9922328837\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9922328837\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 + 2.42T + 2T^{2} \) |
| 5 | \( 1 - 1.62T + 5T^{2} \) |
| 7 | \( 1 - 2.13T + 7T^{2} \) |
| 13 | \( 1 + 5.27T + 13T^{2} \) |
| 17 | \( 1 - 5.69T + 17T^{2} \) |
| 19 | \( 1 + 3.03T + 19T^{2} \) |
| 23 | \( 1 - 4.60T + 23T^{2} \) |
| 29 | \( 1 - 8.32T + 29T^{2} \) |
| 31 | \( 1 - 2.27T + 31T^{2} \) |
| 37 | \( 1 - 0.699T + 37T^{2} \) |
| 41 | \( 1 + 9.03T + 41T^{2} \) |
| 43 | \( 1 + 5.76T + 43T^{2} \) |
| 47 | \( 1 - 2.26T + 47T^{2} \) |
| 53 | \( 1 + 4.65T + 53T^{2} \) |
| 59 | \( 1 + 4.54T + 59T^{2} \) |
| 67 | \( 1 - 4.47T + 67T^{2} \) |
| 71 | \( 1 - 12.9T + 71T^{2} \) |
| 73 | \( 1 - 7.59T + 73T^{2} \) |
| 79 | \( 1 - 8.14T + 79T^{2} \) |
| 83 | \( 1 + 10.6T + 83T^{2} \) |
| 89 | \( 1 - 17.1T + 89T^{2} \) |
| 97 | \( 1 - 11.2T + 97T^{2} \) |
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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
−8.099808857642991965960297486279, −7.68602584201068762003455104757, −6.87784271845791583421227995882, −6.28519405924637790228938736817, −5.18308035466446515811763807986, −4.77002517908242472211003553445, −3.19364882540346299742465888152, −2.32958642114801802198205493660, −1.68394975039067857060837576143, −0.68919258133704829523413410400,
0.68919258133704829523413410400, 1.68394975039067857060837576143, 2.32958642114801802198205493660, 3.19364882540346299742465888152, 4.77002517908242472211003553445, 5.18308035466446515811763807986, 6.28519405924637790228938736817, 6.87784271845791583421227995882, 7.68602584201068762003455104757, 8.099808857642991965960297486279
