| L(s) = 1 | − 0.607·2-s − 2.70·3-s − 1.63·4-s + 1.64·6-s + 4.13·7-s + 2.20·8-s + 4.32·9-s − 2.90·11-s + 4.41·12-s − 5.65·13-s − 2.50·14-s + 1.92·16-s − 5.58·17-s − 2.62·18-s − 4.71·19-s − 11.1·21-s + 1.76·22-s − 0.792·23-s − 5.96·24-s + 3.42·26-s − 3.59·27-s − 6.74·28-s + 6.62·29-s − 9.77·31-s − 5.57·32-s + 7.87·33-s + 3.38·34-s + ⋯ |
| L(s) = 1 | − 0.429·2-s − 1.56·3-s − 0.815·4-s + 0.670·6-s + 1.56·7-s + 0.779·8-s + 1.44·9-s − 0.877·11-s + 1.27·12-s − 1.56·13-s − 0.670·14-s + 0.481·16-s − 1.35·17-s − 0.619·18-s − 1.08·19-s − 2.44·21-s + 0.376·22-s − 0.165·23-s − 1.21·24-s + 0.672·26-s − 0.692·27-s − 1.27·28-s + 1.23·29-s − 1.75·31-s − 0.985·32-s + 1.37·33-s + 0.581·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2354871630\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2354871630\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 + T \) |
| good | 2 | \( 1 + 0.607T + 2T^{2} \) |
| 3 | \( 1 + 2.70T + 3T^{2} \) |
| 7 | \( 1 - 4.13T + 7T^{2} \) |
| 11 | \( 1 + 2.90T + 11T^{2} \) |
| 13 | \( 1 + 5.65T + 13T^{2} \) |
| 17 | \( 1 + 5.58T + 17T^{2} \) |
| 19 | \( 1 + 4.71T + 19T^{2} \) |
| 23 | \( 1 + 0.792T + 23T^{2} \) |
| 29 | \( 1 - 6.62T + 29T^{2} \) |
| 31 | \( 1 + 9.77T + 31T^{2} \) |
| 37 | \( 1 + 0.377T + 37T^{2} \) |
| 41 | \( 1 + 1.27T + 41T^{2} \) |
| 43 | \( 1 - 4.04T + 43T^{2} \) |
| 47 | \( 1 - 8.11T + 47T^{2} \) |
| 53 | \( 1 + 4.86T + 53T^{2} \) |
| 59 | \( 1 + 7.15T + 59T^{2} \) |
| 61 | \( 1 + 14.5T + 61T^{2} \) |
| 67 | \( 1 - 5.68T + 67T^{2} \) |
| 71 | \( 1 + 2.26T + 71T^{2} \) |
| 73 | \( 1 - 15.3T + 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 + 9.54T + 83T^{2} \) |
| 89 | \( 1 + 2.36T + 89T^{2} \) |
| 97 | \( 1 - 12.9T + 97T^{2} \) |
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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
−7.994652950022955593318863719954, −7.45495253946979626743663354078, −6.71868580237014571787620713302, −5.68506275072120228218057353870, −5.12926705257096270992080946873, −4.60047442058016482419741900182, −4.27727269030104972226568317782, −2.43734646665086117219050257049, −1.57324230681862464335693781904, −0.30186959772085349621741932004,
0.30186959772085349621741932004, 1.57324230681862464335693781904, 2.43734646665086117219050257049, 4.27727269030104972226568317782, 4.60047442058016482419741900182, 5.12926705257096270992080946873, 5.68506275072120228218057353870, 6.71868580237014571787620713302, 7.45495253946979626743663354078, 7.994652950022955593318863719954
