| L(s) = 1 | − 1.98·2-s + 1.84·3-s + 1.92·4-s + 2.60·5-s − 3.66·6-s + 2.18·7-s + 0.144·8-s + 0.418·9-s − 5.15·10-s + 3.56·12-s − 0.444·13-s − 4.32·14-s + 4.80·15-s − 4.14·16-s − 6.05·17-s − 0.830·18-s + 1.92·19-s + 5.01·20-s + 4.03·21-s − 9.17·23-s + 0.267·24-s + 1.76·25-s + 0.880·26-s − 4.77·27-s + 4.20·28-s − 10.5·29-s − 9.52·30-s + ⋯ |
| L(s) = 1 | − 1.40·2-s + 1.06·3-s + 0.963·4-s + 1.16·5-s − 1.49·6-s + 0.824·7-s + 0.0511·8-s + 0.139·9-s − 1.62·10-s + 1.02·12-s − 0.123·13-s − 1.15·14-s + 1.24·15-s − 1.03·16-s − 1.46·17-s − 0.195·18-s + 0.440·19-s + 1.12·20-s + 0.880·21-s − 1.91·23-s + 0.0546·24-s + 0.352·25-s + 0.172·26-s − 0.918·27-s + 0.794·28-s − 1.96·29-s − 1.73·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 + 1.98T + 2T^{2} \) |
| 3 | \( 1 - 1.84T + 3T^{2} \) |
| 5 | \( 1 - 2.60T + 5T^{2} \) |
| 7 | \( 1 - 2.18T + 7T^{2} \) |
| 13 | \( 1 + 0.444T + 13T^{2} \) |
| 17 | \( 1 + 6.05T + 17T^{2} \) |
| 19 | \( 1 - 1.92T + 19T^{2} \) |
| 23 | \( 1 + 9.17T + 23T^{2} \) |
| 29 | \( 1 + 10.5T + 29T^{2} \) |
| 31 | \( 1 - 3.13T + 31T^{2} \) |
| 37 | \( 1 + 1.07T + 37T^{2} \) |
| 41 | \( 1 + 1.30T + 41T^{2} \) |
| 43 | \( 1 - 12.5T + 43T^{2} \) |
| 47 | \( 1 + 1.08T + 47T^{2} \) |
| 53 | \( 1 + 4.16T + 53T^{2} \) |
| 59 | \( 1 - 8.24T + 59T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 + 5.20T + 71T^{2} \) |
| 73 | \( 1 - 0.283T + 73T^{2} \) |
| 79 | \( 1 + 14.7T + 79T^{2} \) |
| 83 | \( 1 + 15.2T + 83T^{2} \) |
| 89 | \( 1 - 16.3T + 89T^{2} \) |
| 97 | \( 1 - 4.78T + 97T^{2} \) |
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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
−7.62964417906614174507504991126, −7.41217539287908632882440226859, −6.25009618331047257224858603701, −5.68978645517030911501798076169, −4.62599540005817722563850380685, −3.86998160553829383338588191970, −2.55320610222260791040071957073, −2.04460468237425921678341311137, −1.55668530442005558972167652927, 0,
1.55668530442005558972167652927, 2.04460468237425921678341311137, 2.55320610222260791040071957073, 3.86998160553829383338588191970, 4.62599540005817722563850380685, 5.68978645517030911501798076169, 6.25009618331047257224858603701, 7.41217539287908632882440226859, 7.62964417906614174507504991126
