| L(s) = 1 | − 1.97·2-s + 1.88·4-s − 0.788·5-s − 2.33·7-s + 0.220·8-s + 1.55·10-s + 2.70·11-s + 4.77·13-s + 4.59·14-s − 4.21·16-s + 0.185·17-s − 4.91·19-s − 1.48·20-s − 5.34·22-s − 4.39·23-s − 4.37·25-s − 9.40·26-s − 4.39·28-s + 9.18·29-s + 3.19·31-s + 7.86·32-s − 0.366·34-s + 1.83·35-s − 6.31·37-s + 9.69·38-s − 0.173·40-s + 7.67·41-s + ⋯ |
| L(s) = 1 | − 1.39·2-s + 0.944·4-s − 0.352·5-s − 0.880·7-s + 0.0779·8-s + 0.491·10-s + 0.816·11-s + 1.32·13-s + 1.22·14-s − 1.05·16-s + 0.0451·17-s − 1.12·19-s − 0.332·20-s − 1.13·22-s − 0.915·23-s − 0.875·25-s − 1.84·26-s − 0.831·28-s + 1.70·29-s + 0.574·31-s + 1.38·32-s − 0.0628·34-s + 0.310·35-s − 1.03·37-s + 1.57·38-s − 0.0274·40-s + 1.19·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6684343674\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6684343674\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 149 | \( 1 - T \) |
| good | 2 | \( 1 + 1.97T + 2T^{2} \) |
| 5 | \( 1 + 0.788T + 5T^{2} \) |
| 7 | \( 1 + 2.33T + 7T^{2} \) |
| 11 | \( 1 - 2.70T + 11T^{2} \) |
| 13 | \( 1 - 4.77T + 13T^{2} \) |
| 17 | \( 1 - 0.185T + 17T^{2} \) |
| 19 | \( 1 + 4.91T + 19T^{2} \) |
| 23 | \( 1 + 4.39T + 23T^{2} \) |
| 29 | \( 1 - 9.18T + 29T^{2} \) |
| 31 | \( 1 - 3.19T + 31T^{2} \) |
| 37 | \( 1 + 6.31T + 37T^{2} \) |
| 41 | \( 1 - 7.67T + 41T^{2} \) |
| 43 | \( 1 - 7.21T + 43T^{2} \) |
| 47 | \( 1 - 10.6T + 47T^{2} \) |
| 53 | \( 1 + 4.33T + 53T^{2} \) |
| 59 | \( 1 + 2.18T + 59T^{2} \) |
| 61 | \( 1 - 3.88T + 61T^{2} \) |
| 67 | \( 1 + 1.55T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 - 7.10T + 73T^{2} \) |
| 79 | \( 1 + 15.6T + 79T^{2} \) |
| 83 | \( 1 - 3.09T + 83T^{2} \) |
| 89 | \( 1 - 0.321T + 89T^{2} \) |
| 97 | \( 1 + 12.3T + 97T^{2} \) |
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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
−8.539348520768307920495979123012, −7.985243701164502809688797408776, −7.10362157522718793460343836121, −6.37753421161567348236887648407, −5.95128041312891655940034491734, −4.34876075655898900192374111396, −3.91281606725571337636809272933, −2.73514915686658976272398928116, −1.60657280179897248485829167656, −0.60038338543287298013381873539,
0.60038338543287298013381873539, 1.60657280179897248485829167656, 2.73514915686658976272398928116, 3.91281606725571337636809272933, 4.34876075655898900192374111396, 5.95128041312891655940034491734, 6.37753421161567348236887648407, 7.10362157522718793460343836121, 7.985243701164502809688797408776, 8.539348520768307920495979123012
