| L(s) = 1 | − 0.812·2-s − 0.415·3-s − 1.33·4-s + 2.67·5-s + 0.337·6-s + 2.71·8-s − 2.82·9-s − 2.17·10-s − 0.776·11-s + 0.556·12-s + 3.59·13-s − 1.11·15-s + 0.473·16-s − 0.325·17-s + 2.29·18-s − 19-s − 3.58·20-s + 0.630·22-s + 7.39·23-s − 1.12·24-s + 2.15·25-s − 2.92·26-s + 2.42·27-s − 3.12·29-s + 0.902·30-s − 2.63·31-s − 5.81·32-s + ⋯ |
| L(s) = 1 | − 0.574·2-s − 0.239·3-s − 0.669·4-s + 1.19·5-s + 0.137·6-s + 0.959·8-s − 0.942·9-s − 0.687·10-s − 0.234·11-s + 0.160·12-s + 0.998·13-s − 0.286·15-s + 0.118·16-s − 0.0789·17-s + 0.541·18-s − 0.229·19-s − 0.801·20-s + 0.134·22-s + 1.54·23-s − 0.230·24-s + 0.430·25-s − 0.573·26-s + 0.465·27-s − 0.580·29-s + 0.164·30-s − 0.473·31-s − 1.02·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.067030004\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.067030004\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 0.812T + 2T^{2} \) |
| 3 | \( 1 + 0.415T + 3T^{2} \) |
| 5 | \( 1 - 2.67T + 5T^{2} \) |
| 11 | \( 1 + 0.776T + 11T^{2} \) |
| 13 | \( 1 - 3.59T + 13T^{2} \) |
| 17 | \( 1 + 0.325T + 17T^{2} \) |
| 23 | \( 1 - 7.39T + 23T^{2} \) |
| 29 | \( 1 + 3.12T + 29T^{2} \) |
| 31 | \( 1 + 2.63T + 31T^{2} \) |
| 37 | \( 1 - 5.87T + 37T^{2} \) |
| 41 | \( 1 + 1.62T + 41T^{2} \) |
| 43 | \( 1 - 6.67T + 43T^{2} \) |
| 47 | \( 1 - 2.04T + 47T^{2} \) |
| 53 | \( 1 - 8.92T + 53T^{2} \) |
| 59 | \( 1 + 8.72T + 59T^{2} \) |
| 61 | \( 1 - 7.04T + 61T^{2} \) |
| 67 | \( 1 - 12.5T + 67T^{2} \) |
| 71 | \( 1 - 14.3T + 71T^{2} \) |
| 73 | \( 1 - 13.8T + 73T^{2} \) |
| 79 | \( 1 + 3.88T + 79T^{2} \) |
| 83 | \( 1 + 8.81T + 83T^{2} \) |
| 89 | \( 1 - 5.40T + 89T^{2} \) |
| 97 | \( 1 - 16.5T + 97T^{2} \) |
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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
−9.926150308285342896346821494148, −9.137341544732634075537144185402, −8.712456895033899057711846047769, −7.74006861399166285671884344674, −6.54232403945132202804976331856, −5.66527188076400230208798824939, −5.05735662282648087503047443083, −3.71277490400983914789673850677, −2.33830644632319030203966255852, −0.938818314941439700302789225650,
0.938818314941439700302789225650, 2.33830644632319030203966255852, 3.71277490400983914789673850677, 5.05735662282648087503047443083, 5.66527188076400230208798824939, 6.54232403945132202804976331856, 7.74006861399166285671884344674, 8.712456895033899057711846047769, 9.137341544732634075537144185402, 9.926150308285342896346821494148
