| L(s) = 1 | + 2.31·2-s − 3.27·3-s + 3.37·4-s − 2.64·5-s − 7.59·6-s + 3.57·7-s + 3.19·8-s + 7.73·9-s − 6.14·10-s − 11.0·12-s − 1.34·13-s + 8.28·14-s + 8.67·15-s + 0.650·16-s + 4.94·17-s + 17.9·18-s − 2.80·19-s − 8.94·20-s − 11.7·21-s + 5.42·23-s − 10.4·24-s + 2.01·25-s − 3.12·26-s − 15.5·27-s + 12.0·28-s − 7.47·29-s + 20.1·30-s + ⋯ |
| L(s) = 1 | + 1.63·2-s − 1.89·3-s + 1.68·4-s − 1.18·5-s − 3.10·6-s + 1.34·7-s + 1.12·8-s + 2.57·9-s − 1.94·10-s − 3.19·12-s − 0.374·13-s + 2.21·14-s + 2.24·15-s + 0.162·16-s + 1.19·17-s + 4.23·18-s − 0.642·19-s − 1.99·20-s − 2.55·21-s + 1.13·23-s − 2.13·24-s + 0.402·25-s − 0.613·26-s − 2.98·27-s + 2.27·28-s − 1.38·29-s + 3.67·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.359330088\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.359330088\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 - 2.31T + 2T^{2} \) |
| 3 | \( 1 + 3.27T + 3T^{2} \) |
| 5 | \( 1 + 2.64T + 5T^{2} \) |
| 7 | \( 1 - 3.57T + 7T^{2} \) |
| 13 | \( 1 + 1.34T + 13T^{2} \) |
| 17 | \( 1 - 4.94T + 17T^{2} \) |
| 19 | \( 1 + 2.80T + 19T^{2} \) |
| 23 | \( 1 - 5.42T + 23T^{2} \) |
| 29 | \( 1 + 7.47T + 29T^{2} \) |
| 37 | \( 1 - 8.14T + 37T^{2} \) |
| 41 | \( 1 + 7.74T + 41T^{2} \) |
| 43 | \( 1 - 3.88T + 43T^{2} \) |
| 47 | \( 1 - 2.62T + 47T^{2} \) |
| 53 | \( 1 - 13.4T + 53T^{2} \) |
| 59 | \( 1 + 10.9T + 59T^{2} \) |
| 61 | \( 1 - 4.18T + 61T^{2} \) |
| 67 | \( 1 + 3.62T + 67T^{2} \) |
| 71 | \( 1 - 16.4T + 71T^{2} \) |
| 73 | \( 1 - 11.7T + 73T^{2} \) |
| 79 | \( 1 - 8.85T + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 - 12.2T + 89T^{2} \) |
| 97 | \( 1 + 10.1T + 97T^{2} \) |
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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
−7.983985055980473385561790364205, −7.45729481708827939829471140192, −6.81755160357892770432646241401, −5.93891954229576586194368089531, −5.24168209390950832325335587432, −4.90583401933404494998761260019, −4.17862203147893415279165928555, −3.58441217368351087670016632719, −2.02764363448237512484682611817, −0.78111541940844935886715078239,
0.78111541940844935886715078239, 2.02764363448237512484682611817, 3.58441217368351087670016632719, 4.17862203147893415279165928555, 4.90583401933404494998761260019, 5.24168209390950832325335587432, 5.93891954229576586194368089531, 6.81755160357892770432646241401, 7.45729481708827939829471140192, 7.983985055980473385561790364205
