| L(s) = 1 | + 0.302·2-s − 1.90·4-s − 3.34·5-s + 0.428·7-s − 1.18·8-s − 1.01·10-s − 4.01·11-s + 5.31·13-s + 0.129·14-s + 3.45·16-s − 1.05·17-s − 3.12·19-s + 6.39·20-s − 1.21·22-s − 23-s + 6.22·25-s + 1.60·26-s − 0.817·28-s − 29-s − 10.3·31-s + 3.41·32-s − 0.317·34-s − 1.43·35-s − 10.3·37-s − 0.946·38-s + 3.96·40-s − 7.14·41-s + ⋯ |
| L(s) = 1 | + 0.214·2-s − 0.954·4-s − 1.49·5-s + 0.161·7-s − 0.418·8-s − 0.320·10-s − 1.21·11-s + 1.47·13-s + 0.0346·14-s + 0.864·16-s − 0.254·17-s − 0.716·19-s + 1.42·20-s − 0.259·22-s − 0.208·23-s + 1.24·25-s + 0.315·26-s − 0.154·28-s − 0.185·29-s − 1.86·31-s + 0.603·32-s − 0.0545·34-s − 0.242·35-s − 1.70·37-s − 0.153·38-s + 0.626·40-s − 1.11·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4338631283\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4338631283\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 0.302T + 2T^{2} \) |
| 5 | \( 1 + 3.34T + 5T^{2} \) |
| 7 | \( 1 - 0.428T + 7T^{2} \) |
| 11 | \( 1 + 4.01T + 11T^{2} \) |
| 13 | \( 1 - 5.31T + 13T^{2} \) |
| 17 | \( 1 + 1.05T + 17T^{2} \) |
| 19 | \( 1 + 3.12T + 19T^{2} \) |
| 31 | \( 1 + 10.3T + 31T^{2} \) |
| 37 | \( 1 + 10.3T + 37T^{2} \) |
| 41 | \( 1 + 7.14T + 41T^{2} \) |
| 43 | \( 1 - 3.22T + 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 - 9.70T + 53T^{2} \) |
| 59 | \( 1 - 4.68T + 59T^{2} \) |
| 61 | \( 1 + 4.20T + 61T^{2} \) |
| 67 | \( 1 + 3.89T + 67T^{2} \) |
| 71 | \( 1 + 13.7T + 71T^{2} \) |
| 73 | \( 1 - 8.38T + 73T^{2} \) |
| 79 | \( 1 - 8.80T + 79T^{2} \) |
| 83 | \( 1 + 7.18T + 83T^{2} \) |
| 89 | \( 1 - 10.2T + 89T^{2} \) |
| 97 | \( 1 + 3.31T + 97T^{2} \) |
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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
−8.234525240210883584642508601656, −7.55516193990803436584189447452, −6.74827254149763002087831873067, −5.74188168867290140561133532260, −5.09701704052756637338273931348, −4.36826163265553759728987537338, −3.63539544620577461369598740160, −3.27394025982300348087931174364, −1.76664994867231027760754196342, −0.33209782609367710148423474540,
0.33209782609367710148423474540, 1.76664994867231027760754196342, 3.27394025982300348087931174364, 3.63539544620577461369598740160, 4.36826163265553759728987537338, 5.09701704052756637338273931348, 5.74188168867290140561133532260, 6.74827254149763002087831873067, 7.55516193990803436584189447452, 8.234525240210883584642508601656
