| L(s) = 1 | − 0.396·2-s − 1.26·3-s − 1.84·4-s + 0.441·5-s + 0.502·6-s − 0.667·7-s + 1.52·8-s − 1.39·9-s − 0.175·10-s − 11-s + 2.33·12-s − 1.60·13-s + 0.264·14-s − 0.560·15-s + 3.08·16-s + 4.05·17-s + 0.551·18-s − 0.814·20-s + 0.846·21-s + 0.396·22-s − 0.374·23-s − 1.93·24-s − 4.80·25-s + 0.636·26-s + 5.56·27-s + 1.23·28-s + 2.56·29-s + ⋯ |
| L(s) = 1 | − 0.280·2-s − 0.732·3-s − 0.921·4-s + 0.197·5-s + 0.205·6-s − 0.252·7-s + 0.538·8-s − 0.464·9-s − 0.0553·10-s − 0.301·11-s + 0.674·12-s − 0.445·13-s + 0.0707·14-s − 0.144·15-s + 0.770·16-s + 0.983·17-s + 0.130·18-s − 0.182·20-s + 0.184·21-s + 0.0845·22-s − 0.0781·23-s − 0.394·24-s − 0.960·25-s + 0.124·26-s + 1.07·27-s + 0.232·28-s + 0.475·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3971 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3971 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4902338273\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4902338273\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 0.396T + 2T^{2} \) |
| 3 | \( 1 + 1.26T + 3T^{2} \) |
| 5 | \( 1 - 0.441T + 5T^{2} \) |
| 7 | \( 1 + 0.667T + 7T^{2} \) |
| 13 | \( 1 + 1.60T + 13T^{2} \) |
| 17 | \( 1 - 4.05T + 17T^{2} \) |
| 23 | \( 1 + 0.374T + 23T^{2} \) |
| 29 | \( 1 - 2.56T + 29T^{2} \) |
| 31 | \( 1 + 9.01T + 31T^{2} \) |
| 37 | \( 1 + 7.24T + 37T^{2} \) |
| 41 | \( 1 - 5.08T + 41T^{2} \) |
| 43 | \( 1 + 10.0T + 43T^{2} \) |
| 47 | \( 1 - 2.07T + 47T^{2} \) |
| 53 | \( 1 - 0.326T + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 - 8.12T + 67T^{2} \) |
| 71 | \( 1 + 6.38T + 71T^{2} \) |
| 73 | \( 1 + 0.131T + 73T^{2} \) |
| 79 | \( 1 + 0.269T + 79T^{2} \) |
| 83 | \( 1 + 5.76T + 83T^{2} \) |
| 89 | \( 1 + 0.985T + 89T^{2} \) |
| 97 | \( 1 - 3.37T + 97T^{2} \) |
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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
−8.499455759619994681315468585346, −7.80856939657844648833677409761, −7.04670145229956324557738768367, −6.03928798063504233256550014373, −5.41532354478276761321766168753, −4.95962523099110185918423302597, −3.88234264748826357380064326495, −3.07932697147533834143305337925, −1.73598120307268340635087218015, −0.43589813136329832599549602497,
0.43589813136329832599549602497, 1.73598120307268340635087218015, 3.07932697147533834143305337925, 3.88234264748826357380064326495, 4.95962523099110185918423302597, 5.41532354478276761321766168753, 6.03928798063504233256550014373, 7.04670145229956324557738768367, 7.80856939657844648833677409761, 8.499455759619994681315468585346
